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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_process"> <div class="vector-toc-text"> 2.4 Poisson process </div> </a> <ul id="toc-Poisson_process-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definitions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> 3 Definitions </div> </a> <button aria-controls="toc-Definitions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definitions subsection </button> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-Stochastic_process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stochastic_process"> <div class="vector-toc-text"> 3.1 Stochastic process </div> </a> <ul id="toc-Stochastic_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Index_set" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Index_set"> <div class="vector-toc-text"> 3.2 Index set </div> </a> <ul id="toc-Index_set-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-State_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#State_space"> <div class="vector-toc-text"> 3.3 State space </div> </a> <ul id="toc-State_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sample_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sample_function"> <div class="vector-toc-text"> 3.4 Sample function </div> </a> <ul id="toc-Sample_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Increment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Increment"> <div class="vector-toc-text"> 3.5 Increment </div> </a> <ul id="toc-Increment-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_definitions"> <div class="vector-toc-text"> 3.6 Further definitions </div> </a> <ul id="toc-Further_definitions-sublist" class="vector-toc-list"> <li id="toc-Law" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Law"> <div class="vector-toc-text"> 3.6.1 Law </div> </a> <ul id="toc-Law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite-dimensional_probability_distributions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Finite-dimensional_probability_distributions"> <div class="vector-toc-text"> 3.6.2 Finite-dimensional probability distributions </div> </a> <ul id="toc-Finite-dimensional_probability_distributions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stationarity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Stationarity"> <div class="vector-toc-text"> 3.6.3 Stationarity </div> </a> <ul id="toc-Stationarity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Filtration" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Filtration"> <div class="vector-toc-text"> 3.6.4 Filtration </div> </a> <ul id="toc-Filtration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modification" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modification"> <div class="vector-toc-text"> 3.6.5 Modification </div> </a> <ul id="toc-Modification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Indistinguishable" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Indistinguishable"> <div class="vector-toc-text"> 3.6.6 Indistinguishable </div> </a> <ul id="toc-Indistinguishable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Separability" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Separability"> <div class="vector-toc-text"> 3.6.7 Separability </div> </a> <ul id="toc-Separability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Independence" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Independence"> <div class="vector-toc-text"> 3.6.8 Independence </div> </a> <ul id="toc-Independence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uncorrelatedness" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Uncorrelatedness"> <div class="vector-toc-text"> 3.6.9 Uncorrelatedness </div> </a> <ul id="toc-Uncorrelatedness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Independence_implies_uncorrelatedness" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Independence_implies_uncorrelatedness"> <div class="vector-toc-text"> 3.6.10 Independence implies uncorrelatedness </div> </a> <ul id="toc-Independence_implies_uncorrelatedness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orthogonality" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Orthogonality"> <div class="vector-toc-text"> 3.6.11 Orthogonality </div> </a> <ul id="toc-Orthogonality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skorokhod_space" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Skorokhod_space"> <div class="vector-toc-text"> 3.6.12 Skorokhod space </div> </a> <ul id="toc-Skorokhod_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Regularity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Regularity"> <div class="vector-toc-text"> 3.6.13 Regularity </div> </a> <ul id="toc-Regularity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Further_examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_examples"> <div class="vector-toc-text"> 4 Further examples </div> </a> <button aria-controls="toc-Further_examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Further examples subsection </button> <ul id="toc-Further_examples-sublist" class="vector-toc-list"> <li id="toc-Markov_processes_and_chains" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Markov_processes_and_chains"> <div class="vector-toc-text"> 4.1 Markov processes and chains </div> </a> <ul id="toc-Markov_processes_and_chains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Martingale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Martingale"> <div class="vector-toc-text"> 4.2 Martingale </div> </a> <ul id="toc-Martingale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lévy_process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lévy_process"> <div class="vector-toc-text"> 4.3 Lévy process </div> </a> <ul id="toc-Lévy_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Random_field"> <div class="vector-toc-text"> 4.4 Random field </div> </a> <ul id="toc-Random_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Point_process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Point_process"> <div class="vector-toc-text"> 4.5 Point process </div> </a> <ul id="toc-Point_process-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 5 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Early_probability_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Early_probability_theory"> <div class="vector-toc-text"> 5.1 Early probability theory </div> </a> <ul id="toc-Early_probability_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistical_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistical_mechanics"> <div class="vector-toc-text"> 5.2 Statistical mechanics </div> </a> <ul id="toc-Statistical_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measure_theory_and_probability_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measure_theory_and_probability_theory"> <div class="vector-toc-text"> 5.3 Measure theory and probability theory </div> </a> <ul id="toc-Measure_theory_and_probability_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Birth_of_modern_probability_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Birth_of_modern_probability_theory"> <div class="vector-toc-text"> 5.4 Birth of modern probability theory </div> </a> <ul id="toc-Birth_of_modern_probability_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stochastic_processes_after_World_War_II" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stochastic_processes_after_World_War_II"> <div class="vector-toc-text"> 5.5 Stochastic processes after World War II </div> </a> <ul id="toc-Stochastic_processes_after_World_War_II-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discoveries_of_specific_stochastic_processes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discoveries_of_specific_stochastic_processes"> <div class="vector-toc-text"> 5.6 Discoveries of specific stochastic processes </div> </a> <ul id="toc-Discoveries_of_specific_stochastic_processes-sublist" class="vector-toc-list"> <li id="toc-Bernoulli_process_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bernoulli_process_2"> <div class="vector-toc-text"> 5.6.1 Bernoulli process </div> </a> <ul id="toc-Bernoulli_process_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_walks" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Random_walks"> <div class="vector-toc-text"> 5.6.2 Random walks </div> </a> <ul id="toc-Random_walks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wiener_process_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Wiener_process_2"> <div class="vector-toc-text"> 5.6.3 Wiener process </div> </a> <ul id="toc-Wiener_process_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_process_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Poisson_process_2"> <div class="vector-toc-text"> 5.6.4 Poisson process </div> </a> <ul id="toc-Poisson_process_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Markov_processes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Markov_processes"> <div class="vector-toc-text"> 5.6.5 Markov processes </div> </a> <ul id="toc-Markov_processes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lévy_processes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lévy_processes"> <div class="vector-toc-text"> 5.6.6 Lévy processes </div> </a> <ul id="toc-Lévy_processes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Mathematical_construction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mathematical_construction"> <div class="vector-toc-text"> 6 Mathematical construction </div> </a> <button aria-controls="toc-Mathematical_construction-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Mathematical construction subsection </button> <ul id="toc-Mathematical_construction-sublist" class="vector-toc-list"> <li id="toc-Construction_issues" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction_issues"> <div class="vector-toc-text"> 6.1 Construction issues </div> </a> <ul id="toc-Construction_issues-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resolving_construction_issues" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Resolving_construction_issues"> <div class="vector-toc-text"> 6.2 Resolving construction issues </div> </a> <ul id="toc-Resolving_construction_issues-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Application" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Application"> <div class="vector-toc-text"> 7 Application </div> </a> <button aria-controls="toc-Application-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Application subsection </button> <ul id="toc-Application-sublist" class="vector-toc-list"> <li id="toc-Applications_in_Finance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications_in_Finance"> <div class="vector-toc-text"> 7.1 Applications in Finance </div> </a> <ul id="toc-Applications_in_Finance-sublist" class="vector-toc-list"> <li id="toc-Black-Scholes_Model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Black-Scholes_Model"> <div class="vector-toc-text"> 7.1.1 Black-Scholes Model </div> </a> <ul id="toc-Black-Scholes_Model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stochastic_Volatility_Models" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Stochastic_Volatility_Models"> <div class="vector-toc-text"> 7.1.2 Stochastic Volatility Models </div> </a> <ul id="toc-Stochastic_Volatility_Models-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications_in_Biology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications_in_Biology"> <div class="vector-toc-text"> 7.2 Applications in Biology </div> </a> <ul id="toc-Applications_in_Biology-sublist" class="vector-toc-list"> <li id="toc-Population_Dynamics" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Population_Dynamics"> <div class="vector-toc-text"> 7.2.1 Population Dynamics </div> </a> <ul id="toc-Population_Dynamics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 10 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 11 Further reading </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Further reading subsection </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Articles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles"> <div class="vector-toc-text"> 11.1 Articles </div> </a> <ul id="toc-Articles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books"> <div class="vector-toc-text"> 11.2 Books </div> </a> <ul id="toc-Books-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 12 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown 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Available in 46 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-46" aria-hidden="true" > 46 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Stogastiese_proses" title="Stogastiese proses – Afrikaans" lang="af" hreflang="af" data-title="Stogastiese proses" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D9%85%D9%84%D9%8A%D8%A9_%D8%AA%D8%B5%D8%A7%D8%AF%D9%81%D9%8A%D8%A9" title="عملية تصادفية – Arabic" lang="ar" hreflang="ar" data-title="عملية تصادفية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Procesu_estoc%C3%A1sticu" title="Procesu estocásticu – Asturian" lang="ast" hreflang="ast" data-title="Procesu estocásticu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A7%88%E0%A6%AC_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE" title="দৈব প্রক্রিয়া – Bangla" lang="bn" hreflang="bn" data-title="দৈব প্রক্রিয়া" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%82%D0%BE%D1%85%D0%B0%D1%81%D1%82%D0%B8%D1%87%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81" title="Стохастичен процес – Bulgarian" lang="bg" hreflang="bg" data-title="Стохастичен процес" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Proc%C3%A9s_estoc%C3%A0stic" title="Procés estocàstic – Catalan" lang="ca" hreflang="ca" data-title="Procés estocàstic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/N%C3%A1hodn%C3%BD_proces" title="Náhodný proces – Czech" lang="cs" hreflang="cs" data-title="Náhodný proces" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Stochastischer_Prozess" title="Stochastischer Prozess – German" lang="de" hreflang="de" data-title="Stochastischer Prozess" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Juhuslik_protsess" title="Juhuslik protsess – Estonian" lang="et" hreflang="et" data-title="Juhuslik protsess" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Proceso_estoc%C3%A1stico" title="Proceso estocástico – Spanish" lang="es" hreflang="es" data-title="Proceso estocástico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Stokastiko" title="Stokastiko – Esperanto" lang="eo" hreflang="eo" data-title="Stokastiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Prozesu_estokastiko" title="Prozesu estokastiko – Basque" lang="eu" hreflang="eu" data-title="Prozesu estokastiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B1%D8%A7%DB%8C%D9%86%D8%AF_%D8%AA%D8%B5%D8%A7%D8%AF%D9%81%DB%8C" title="فرایند تصادفی – Persian" lang="fa" hreflang="fa" data-title="فرایند تصادفی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Processus_stochastique" title="Processus stochastique – French" lang="fr" hreflang="fr" data-title="Processus stochastique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Pr%C3%B3iseas_stocastach" title="Próiseas stocastach – Irish" lang="ga" hreflang="ga" data-title="Próiseas stocastach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Proceso_estoc%C3%A1stico" title="Proceso estocástico – Galician" lang="gl" hreflang="gl" data-title="Proceso estocástico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%95%EB%A5%A0_%EA%B3%BC%EC%A0%95" title="확률 과정 – Korean" lang="ko" hreflang="ko" data-title="확률 과정" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%A1%D5%BF%D5%A1%D5%B0%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BA%D6%80%D5%B8%D6%81%D5%A5%D5%BD" title="Պատահական պրոցես – Armenian" lang="hy" hreflang="hy" data-title="Պատահական պրոցես" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B8%E0%A4%AE%E0%A5%8D%E0%A4%AD%E0%A4%BE%E0%A4%B5%E0%A5%8D%E2%80%8D%E0%A4%AF_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE" title="प्रसम्भाव्‍य प्रक्रम – Hindi" lang="hi" hreflang="hi" data-title="प्रसम्भाव्‍य प्रक्रम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Proses_stokastik" title="Proses stokastik – Indonesian" lang="id" hreflang="id" data-title="Proses stokastik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Processo_stocastico" title="Processo stocastico – Italian" lang="it" hreflang="it" data-title="Processo stocastico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%94%D7%9C%D7%99%D7%9A_%D7%A1%D7%98%D7%95%D7%9B%D7%A1%D7%98%D7%99" title="תהליך סטוכסטי – Hebrew" lang="he" hreflang="he" data-title="תהליך סטוכסטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B5%D0%B7%D0%B4%D0%B5%D0%B9%D1%81%D0%BE%D2%9B_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81" title="Кездейсоқ процесс – Kazakh" lang="kk" hreflang="kk" data-title="Кездейсоқ процесс" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sztochasztikus_folyamat" title="Sztochasztikus folyamat – Hungarian" lang="hu" hreflang="hu" data-title="Sztochasztikus folyamat" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B4%AE%E0%B4%BF%E0%B4%B2%E0%B5%8D%E0%B4%B2%E0%B4%BE%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%AF" title="ക്രമമില്ലാപ്രക്രിയ – Malayalam" lang="ml" hreflang="ml" data-title="ക്രമമില്ലാപ്രക്രിയ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Proses_stokastik" title="Proses stokastik – Malay" lang="ms" hreflang="ms" data-title="Proses stokastik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stochastisch_proces" title="Stochastisch proces – Dutch" lang="nl" hreflang="nl" data-title="Stochastisch proces" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A2%BA%E7%8E%87%E9%81%8E%E7%A8%8B" title="確率過程 – Japanese" lang="ja" hreflang="ja" data-title="確率過程" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Stokastisk_prosess" title="Stokastisk prosess – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Stokastisk prosess" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Proces_stochastyczny" title="Proces stochastyczny – Polish" lang="pl" hreflang="pl" data-title="Proces stochastyczny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Processo_estoc%C3%A1stico" title="Processo estocástico – Portuguese" lang="pt" hreflang="pt" data-title="Processo estocástico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Proces_stohastic" title="Proces stohastic – Romanian" lang="ro" hreflang="ro" data-title="Proces stohastic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BB%D1%83%D1%87%D0%B0%D0%B9%D0%BD%D1%8B%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81%D1%81" title="Случайный процесс – Russian" lang="ru" hreflang="ru" data-title="Случайный процесс" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Proceset_stokastike" title="Proceset stokastike – Albanian" lang="sq" hreflang="sq" data-title="Proceset stokastike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Stochastic_process" title="Stochastic process – Simple English" lang="en-simple" hreflang="en-simple" data-title="Stochastic process" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Stochastick%C3%BD_proces" title="Stochastický proces – Slovak" lang="sk" hreflang="sk" data-title="Stochastický proces" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Stohasti%C4%8Dki_proces" title="Stohastički proces – Serbian" lang="sr" hreflang="sr" data-title="Stohastički proces" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Pros%C3%A9s_stokastik" title="Prosés stokastik – Sundanese" lang="su" hreflang="su" data-title="Prosés stokastik" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Stokastinen_prosessi" title="Stokastinen prosessi – Finnish" lang="fi" hreflang="fi" data-title="Stokastinen prosessi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Stokastisk_process" title="Stokastisk process – Swedish" lang="sv" hreflang="sv" data-title="Stokastisk process" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%A7%E0%B8%99%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%9F%E0%B9%89%E0%B8%99%E0%B8%AA%E0%B8%B8%E0%B9%88%E0%B8%A1" title="กระบวนการเฟ้นสุ่ม – Thai" lang="th" hreflang="th" data-title="กระบวนการเฟ้นสุ่ม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Stokastik_s%C3%BCre%C3%A7" title="Stokastik süreç – Turkish" lang="tr" hreflang="tr" data-title="Stokastik süreç" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B8%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%86%D0%B5%D1%81" title="Випадковий процес – Ukrainian" lang="uk" hreflang="uk" data-title="Випадковий процес" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Qu%C3%A1_tr%C3%ACnh_ng%E1%BA%ABu_nhi%C3%AAn" title="Quá trình ngẫu nhiên – Vietnamese" lang="vi" hreflang="vi" data-title="Quá trình ngẫu nhiên" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9A%A8%E6%A9%9F%E9%81%8E%E7%A8%8B" title="隨機過程 – Cantonese" lang="yue" hreflang="yue" data-title="隨機過程" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9A%8F%E6%9C%BA%E8%BF%87%E7%A8%8B" title="随机过程 – Chinese" lang="zh" hreflang="zh" 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process">Wiener</a> or <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a><a href="#cite_note-RogersWilliams2000page1-2">[2]</a><a href="#cite_note-Steele2012page29-3">[3]</a></figcaption></figure> In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> and related fields, a stochastic (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/stəˈkæstɪk/</a>) or random process is a <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> usually defined as a <a href="/wiki/Indexed_family" title="Indexed family">family</a> of <a href="/wiki/Random_variable" title="Random variable">random variables</a> in a <a href="/wiki/Probability_space" title="Probability space">probability space</a>, where the <a href="/wiki/Index_set" title="Index set">index</a> of the family often has the interpretation of <a href="/wiki/Time" title="Time">time</a>. <a href="/wiki/Stochastic" title="Stochastic">Stochastic</a> processes are widely used as <a href="/wiki/Mathematical_model" title="Mathematical model">mathematical models</a> of systems and phenomena that appear to vary in a random manner. Examples include the growth of a <a href="/wiki/Bacteria" title="Bacteria">bacterial</a> population, an <a href="/wiki/Electrical_current" class="mw-redirect" title="Electrical current">electrical current</a> fluctuating due to <a href="/wiki/Thermal_noise" class="mw-redirect" title="Thermal noise">thermal noise</a>, or the movement of a <a href="/wiki/Gas" title="Gas">gas</a> <a href="/wiki/Molecule" title="Molecule">molecule</a>.<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a><a href="#cite_note-Parzen1999-4">[4]</a><a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a> Stochastic processes have applications in many disciplines such as <a href="/wiki/Biology" title="Biology">biology</a>,<a href="#cite_note-Bressloff2014-6">[6]</a> <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>,<a href="#cite_note-Kampen2011-7">[7]</a> <a href="/wiki/Ecology" title="Ecology">ecology</a>,<a href="#cite_note-LandeEngen2003-8">[8]</a> <a href="/wiki/Neuroscience" title="Neuroscience">neuroscience</a>,<a href="#cite_note-LaingLord2010-9">[9]</a> <a href="/wiki/Physics" title="Physics">physics</a>,<a href="#cite_note-PaulBaschnagel2013-10">[10]</a> <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>,<a href="#cite_note-Dougherty1999-11">[11]</a> <a href="/wiki/Stochastic_control" title="Stochastic control">control theory</a>,<a href="#cite_note-Bertsekas1996-12">[12]</a> <a href="/wiki/Information_theory" title="Information theory">information theory</a>,<a href="#cite_note-CoverThomas2012page71-13">[13]</a> <a href="/wiki/Computer_science" title="Computer science">computer science</a>,<a href="#cite_note-Baron2015-14">[14]</a> and <a href="/wiki/Telecommunications" title="Telecommunications">telecommunications</a>.<a href="#cite_note-BaccelliBlaszczyszyn2009-15">[15]</a> Furthermore, seemingly random changes in <a href="/wiki/Financial_market" title="Financial market">financial markets</a> have motivated the extensive use of stochastic processes in <a href="/wiki/Finance" title="Finance">finance</a>.<a href="#cite_note-Steele2001-16">[16]</a><a href="#cite_note-MusielaRutkowski2006-17">[17]</a><a href="#cite_note-Shreve2004-18">[18]</a> Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the <a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a> or Brownian motion process,<a href="#cite_note-21">[a]</a> used by <a href="/wiki/Louis_Bachelier" title="Louis Bachelier">Louis Bachelier</a> to study price changes on the <a href="/wiki/Paris_Bourse" class="mw-redirect" title="Paris Bourse">Paris Bourse</a>,<a href="#cite_note-JarrowProtter2004-22">[21]</a> and the <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a>, used by <a href="/wiki/A._K._Erlang" class="mw-redirect" title="A. K. Erlang">A. K. Erlang</a> to study the number of phone calls occurring in a certain period of time.<a href="#cite_note-Stirzaker2000-23">[22]</a> These two stochastic processes are considered the most important and central in the theory of stochastic processes,<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a><a href="#cite_note-Parzen1999-4">[4]</a><a href="#cite_note-24">[23]</a> and were invented repeatedly and independently, both before and after Bachelier and Erlang, in different settings and countries.<a href="#cite_note-JarrowProtter2004-22">[21]</a><a href="#cite_note-GuttorpThorarinsdottir2012-25">[24]</a> The term random function is also used to refer to a stochastic or random process,<a href="#cite_note-GusakKukush2010page21-26">[25]</a><a href="#cite_note-Skorokhod2005page42-27">[26]</a> because a stochastic process can also be interpreted as a random element in a <a href="/wiki/Function_space" title="Function space">function space</a>.<a href="#cite_note-Kallenberg2002page24-28">[27]</a><a href="#cite_note-Lamperti1977page1-29">[28]</a> The terms stochastic process and random process are used interchangeably, often with no specific <a href="/wiki/Mathematical_space" class="mw-redirect" title="Mathematical space">mathematical space</a> for the set that indexes the random variables.<a href="#cite_note-Kallenberg2002page24-28">[27]</a><a href="#cite_note-ChaumontYor2012-30">[29]</a> But often these two terms are used when the random variables are indexed by the <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a> or an <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> of the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>.<a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-ChaumontYor2012-30">[29]</a> If the random variables are indexed by the <a href="/wiki/Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a> or some higher-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, then the collection of random variables is usually called a <a href="/wiki/Random_field" title="Random field">random field</a> instead.<a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-AdlerTaylor2009page7-31">[30]</a> The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.<a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-Lamperti1977page1-29">[28]</a> Based on their mathematical properties, stochastic processes can be grouped into various categories, which include <a href="/wiki/Random_walk" title="Random walk">random walks</a>,<a href="#cite_note-LawlerLimic2010-32">[31]</a> <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingales</a>,<a href="#cite_note-Williams1991-33">[32]</a> <a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov processes</a>,<a href="#cite_note-RogersWilliams2000-34">[33]</a> <a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy processes</a>,<a href="#cite_note-ApplebaumBook2004-35">[34]</a> <a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian processes</a>,<a href="#cite_note-36">[35]</a> random fields,<a href="#cite_note-Adler2010-37">[36]</a> <a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">renewal processes</a>, and <a href="/wiki/Branching_process" title="Branching process">branching processes</a>.<a href="#cite_note-KarlinTaylor2012-38">[37]</a> The study of stochastic processes uses mathematical knowledge and techniques from <a href="/wiki/Probability" title="Probability">probability</a>, <a href="/wiki/Calculus" title="Calculus">calculus</a>, <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and <a href="/wiki/Topology" title="Topology">topology</a><a href="#cite_note-Hajek2015-39">[38]</a><a href="#cite_note-LatoucheRamaswami1999-40">[39]</a><a href="#cite_note-DaleyVere-Jones2007-41">[40]</a> as well as branches of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> such as <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>, <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, and <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>.<a href="#cite_note-Billingsley2008-42">[41]</a><a href="#cite_note-Brémaud2014-43">[42]</a><a href="#cite_note-Bobrowski2005-44">[43]</a> The theory of stochastic processes is considered to be an important contribution to mathematics<a href="#cite_note-Applebaum2004-45">[44]</a> and it continues to be an active topic of research for both theoretical reasons and applications.<a href="#cite_note-BlathImkeller2011-46">[45]</a><a href="#cite_note-Talagrand2014-47">[46]</a><a href="#cite_note-Bressloff2014VII-48">[47]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=1" title="Edit section: Introduction">edit</a>]</div> A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.<a href="#cite_note-Parzen1999-4">[4]</a><a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a> The set used to index the random variables is called the <a href="/wiki/Index_set" title="Index set">index set</a>. Historically, the index set was some <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>, such as the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>, giving the index set the interpretation of time.<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a> Each random variable in the collection takes values from the same <a href="/wiki/Mathematical_space" class="mw-redirect" title="Mathematical space">mathematical space</a> known as the state space. This state space can be, for example, the integers, the real line or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space.<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a><a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a> An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time.<a href="#cite_note-KarlinTaylor2012page27-49">[48]</a><a href="#cite_note-Applebaum2004page1337-50">[49]</a> A stochastic process can have many <a href="/wiki/Outcome_(probability)" title="Outcome (probability)">outcomes</a>, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization.<a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-RogersWilliams2000page121b-51">[50]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wiener_process_3d.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiener_process_3d.png/220px-Wiener_process_3d.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiener_process_3d.png/330px-Wiener_process_3d.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiener_process_3d.png/440px-Wiener_process_3d.png 2x" data-file-width="904" data-file-height="883" /></a><figcaption>A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Classifications">Classifications</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=2" title="Edit section: Classifications">edit</a>]</div> A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the index set and the state space.<a href="#cite_note-Florescu2014page294-52">[51]</a><a href="#cite_note-KarlinTaylor2012page26-53">[52]</a><a href="#cite_note-54">[53]</a> When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in <a href="/wiki/Discrete_time" class="mw-redirect" title="Discrete time">discrete time</a>.<a href="#cite_note-Billingsley2008page482-55">[54]</a><a href="#cite_note-Borovkov2013page527-56">[55]</a> If the index set is some interval of the real line, then time is said to be <a href="/wiki/Continuous_time" class="mw-redirect" title="Continuous time">continuous</a>. The two types of stochastic processes are respectively referred to as discrete-time and <a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">continuous-time stochastic processes</a>.<a href="#cite_note-KarlinTaylor2012page27-49">[48]</a><a href="#cite_note-Brémaud2014page120-57">[56]</a><a href="#cite_note-Rosenthal2006page177-58">[57]</a> Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.<a href="#cite_note-KloedenPlaten2013page63-59">[58]</a><a href="#cite_note-Khoshnevisan2006page153-60">[59]</a> If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence.<a href="#cite_note-Borovkov2013page527-56">[55]</a> If the state space is the integers or natural numbers, then the stochastic process is called a discrete or integer-valued stochastic process. If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space, then the stochastic process is called a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional vector process or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-vector process.<a href="#cite_note-Florescu2014page294-52">[51]</a><a href="#cite_note-KarlinTaylor2012page26-53">[52]</a> <div class="mw-heading mw-heading3"><h3 id="Etymology">Etymology</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=3" title="Edit section: Etymology">edit</a>]</div> The word stochastic in <a href="/wiki/English_language" title="English language">English</a> was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a <a href="/wiki/Greek_language" title="Greek language">Greek</a> word meaning "to aim at a mark, guess", and the <a href="/wiki/Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a> gives the year 1662 as its earliest occurrence.<a href="#cite_note-OxfordStochastic-61">[60]</a> In his work on probability Ars Conjectandi, originally published in Latin in 1713, <a href="/wiki/Jakob_Bernoulli" class="mw-redirect" title="Jakob Bernoulli">Jakob Bernoulli</a> used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics".<a href="#cite_note-Sheĭnin2006page5-62">[61]</a> This phrase was used, with reference to Bernoulli, by <a href="/wiki/Ladislaus_Bortkiewicz" title="Ladislaus Bortkiewicz">Ladislaus Bortkiewicz</a><a href="#cite_note-SheyninStrecker2011page136-63">[62]</a> who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by <a href="/wiki/Joseph_Doob" class="mw-redirect" title="Joseph Doob">Joseph Doob</a>.<a href="#cite_note-OxfordStochastic-61">[60]</a> For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by <a href="/wiki/Aleksandr_Khinchin" title="Aleksandr Khinchin">Aleksandr Khinchin</a>,<a href="#cite_note-Doob1934-64">[63]</a><a href="#cite_note-Khintchine1934-65">[64]</a> though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.<a href="#cite_note-Kolmogoroff1931page1-66">[65]</a> According to the Oxford English Dictionary, early occurrences of the word random in English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term random process pre-dates stochastic process, which the Oxford English Dictionary also gives as a synonym, and was used in an article by <a href="/wiki/Francis_Edgeworth" class="mw-redirect" title="Francis Edgeworth">Francis Edgeworth</a> published in 1888.<a href="#cite_note-OxfordRandom-67">[66]</a> <div class="mw-heading mw-heading3"><h3 id="Terminology">Terminology</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=4" title="Edit section: Terminology">edit</a>]</div> The definition of a stochastic process varies,<a href="#cite_note-FristedtGray2013page580-68">[67]</a> but a stochastic process is traditionally defined as a collection of random variables indexed by some set.<a href="#cite_note-RogersWilliams2000page121-69">[68]</a><a href="#cite_note-Asmussen2003page408-70">[69]</a> The terms random process and stochastic process are considered synonyms and are used interchangeably, without the index set being precisely specified.<a href="#cite_note-Kallenberg2002page24-28">[27]</a><a href="#cite_note-ChaumontYor2012-30">[29]</a><a href="#cite_note-AdlerTaylor2009page7-31">[30]</a><a href="#cite_note-Stirzaker2005page45-71">[70]</a><a href="#cite_note-Rosenblatt1962page91-72">[71]</a><a href="#cite_note-Gubner2006page383-73">[72]</a> Both "collection",<a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-Stirzaker2005page45-71">[70]</a> or "family" are used<a href="#cite_note-Parzen1999-4">[4]</a><a href="#cite_note-Ito2006page13-74">[73]</a> while instead of "index set", sometimes the terms "parameter set"<a href="#cite_note-Lamperti1977page1-29">[28]</a> or "parameter space"<a href="#cite_note-AdlerTaylor2009page7-31">[30]</a> are used. The term random function is also used to refer to a stochastic or random process,<a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-Loeve1978-75">[74]</a><a href="#cite_note-Brémaud2014page133-76">[75]</a> though sometimes it is only used when the stochastic process takes real values.<a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-Ito2006page13-74">[73]</a> This term is also used when the index sets are mathematical spaces other than the real line,<a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-GusakKukush2010page1-77">[76]</a> while the terms stochastic process and random process are usually used when the index set is interpreted as time,<a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-GusakKukush2010page1-77">[76]</a><a href="#cite_note-Bass2011page1-78">[77]</a> and other terms are used such as random field when the index set is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> or a <a href="/wiki/Manifold" title="Manifold">manifold</a>.<a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-AdlerTaylor2009page7-31">[30]</a> <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=5" title="Edit section: Notation">edit</a>]</div> A stochastic process can be denoted, among other ways, by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X(t)\}_{t\in T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X(t)\}_{t\in T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15bf3d3e7f4ec8cf9df7761e60f7cc5f103fb8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.033ex; height:2.843ex;" alt="{\displaystyle \{X(t)\}_{t\in T}}">,<a href="#cite_note-Brémaud2014page120-57">[56]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{t}\}_{t\in T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\in T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18d29bc7ff1908644cc092f09185407eb6391d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.154ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\in T}}">,<a href="#cite_note-Asmussen2003page408-70">[69]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{t}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aa498a302423f64556e0783cd198ee7def541bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.075ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}}"><a href="#cite_note-Lamperti1977page3-79">[78]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X(t)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X(t)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83c817aa0d6995481747222c20ae8dceb374929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.954ex; height:2.843ex;" alt="{\displaystyle \{X(t)\}}"> or simply as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">. Some authors mistakenly write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfeb6663c0a903f587cd6d776c387370fc5c4ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.629ex; height:2.843ex;" alt="{\displaystyle X(t)}"> even though it is an <a href="/wiki/Abuse_of_notation#Function_notation" title="Abuse of notation">abuse of function notation</a>.<a href="#cite_note-Klebaner2005page55-80">[79]</a> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfeb6663c0a903f587cd6d776c387370fc5c4ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.629ex; height:2.843ex;" alt="{\displaystyle X(t)}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}"> are used to refer to the random variable with the index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">, and not the entire stochastic process.<a href="#cite_note-Lamperti1977page3-79">[78]</a> If the index set is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=[0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=[0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d5792c68149c5c2f3e434e8c119a702ced9846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.806ex; height:2.843ex;" alt="{\displaystyle T=[0,\infty )}">, then one can write, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{t},t\geq 0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X_{t},t\geq 0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a71cad170c67eba2ce9d1b02bd336353941a3f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.694ex; height:2.843ex;" alt="{\displaystyle (X_{t},t\geq 0)}"> to denote the stochastic process.<a href="#cite_note-ChaumontYor2012-30">[29]</a> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=6" title="Edit section: Examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Bernoulli_process">Bernoulli process</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=7" title="Edit section: Bernoulli process">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></div> One of the simplest stochastic processes is the <a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a>,<a href="#cite_note-Florescu2014page293-81">[80]</a> which is a sequence of <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> (iid) random variables, where each random variable takes either the value one or zero, say one with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> and zero with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9633a8692121eedfa99cace406205e5d1511ef8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}">. This process can be linked to an idealisation of repeatedly flipping a coin, where the probability of obtaining a head is taken to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> and its value is one, while the value of a tail is zero.<a href="#cite_note-Florescu2014page301-82">[81]</a> In other words, a Bernoulli process is a sequence of iid Bernoulli random variables,<a href="#cite_note-BertsekasTsitsiklis2002page273-83">[82]</a> where each idealised coin flip is an example of a <a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a>.<a href="#cite_note-Ibe2013page11-84">[83]</a> <div class="mw-heading mw-heading3"><h3 id="Random_walk">Random walk</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=8" title="Edit section: Random walk">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Random_walk" title="Random walk">Random walk</a></div> <a href="/wiki/Random_walks" class="mw-redirect" title="Random walks">Random walks</a> are stochastic processes that are usually defined as sums of <a href="/wiki/Iid" class="mw-redirect" title="Iid">iid</a> random variables or random vectors in Euclidean space, so they are processes that change in discrete time.<a href="#cite_note-Klenke2013page347-85">[84]</a><a href="#cite_note-LawlerLimic2010page1-86">[85]</a><a href="#cite_note-Kallenberg2002page136-87">[86]</a><a href="#cite_note-Florescu2014page383-88">[87]</a><a href="#cite_note-Durrett2010page277-89">[88]</a> But some also use the term to refer to processes that change in continuous time,<a href="#cite_note-Weiss2006page1-90">[89]</a> particularly the Wiener process used in financial models, which has led to some confusion, resulting in its criticism.<a href="#cite_note-Spanos1999page454-91">[90]</a> There are various other types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.<a href="#cite_note-Weiss2006page1-90">[89]</a><a href="#cite_note-Klebaner2005page81-92">[91]</a> A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">, or decreases by one with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9633a8692121eedfa99cace406205e5d1511ef8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}">, so the index set of this random walk is the natural numbers, while its state space is the integers. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=0.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78145978df11bf657030540a5136bda6646bd3ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.329ex; height:2.509ex;" alt="{\displaystyle p=0.5}">, this random walk is called a symmetric random walk.<a href="#cite_note-Gut2012page88-93">[92]</a><a href="#cite_note-GrimmettStirzaker2001page71-94">[93]</a> <div class="mw-heading mw-heading3"><h3 id="Wiener_process">Wiener process</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=9" title="Edit section: Wiener process">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></div> The Wiener process is a stochastic process with <a href="/wiki/Stationary_increments" title="Stationary increments">stationary</a> and <a href="/wiki/Independent_increments" title="Independent increments">independent increments</a> that are <a href="/wiki/Normally_distributed" class="mw-redirect" title="Normally distributed">normally distributed</a> based on the size of the increments.<a href="#cite_note-RogersWilliams2000page1-2">[2]</a><a href="#cite_note-Klebaner2005page56-95">[94]</a> The Wiener process is named after <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a>, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for <a href="/wiki/Brownian_movement" class="mw-redirect" title="Brownian movement">Brownian movement</a> in liquids.<a href="#cite_note-Brush1968page1-96">[95]</a><a href="#cite_note-Applebaum2004page1338-97">[96]</a><a href="#cite_note-GikhmanSkorokhod1969page21-98">[97]</a> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:DriftedWienerProcess1D.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/DriftedWienerProcess1D.svg/220px-DriftedWienerProcess1D.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/DriftedWienerProcess1D.svg/330px-DriftedWienerProcess1D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/DriftedWienerProcess1D.svg/440px-DriftedWienerProcess1D.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption>Realizations of Wiener processes (or Brownian motion processes) with drift (<style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style>blue) and without drift (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494">red)</figcaption></figure> Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes.<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a><a href="#cite_note-RogersWilliams2000page1-2">[2]</a><a href="#cite_note-Steele2012page29-3">[3]</a><a href="#cite_note-Florescu2014page471-99">[98]</a><a href="#cite_note-KarlinTaylor2012page21-100">[99]</a><a href="#cite_note-KaratzasShreve2014pageVIII-101">[100]</a><a href="#cite_note-RevuzYor2013pageIX-102">[101]</a> Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space.<a href="#cite_note-Rosenthal2006page186-103">[102]</a> But the process can be defined more generally so its state space can be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space.<a href="#cite_note-Klebaner2005page81-92">[91]</a><a href="#cite_note-KarlinTaylor2012page21-100">[99]</a><a href="#cite_note-104">[103]</a> If the <a href="/wiki/Mean" title="Mean">mean</a> of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }">, which is a real number, then the resulting stochastic process is said to have drift <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }">.<a href="#cite_note-Steele2012page118-105">[104]</a><a href="#cite_note-MörtersPeres2010page1-106">[105]</a><a href="#cite_note-KaratzasShreve2014page78-107">[106]</a> <a href="/wiki/Almost_surely" title="Almost surely">Almost surely</a>, a sample path of a Wiener process is continuous everywhere but <a href="/wiki/Nowhere_differentiable_function" class="mw-redirect" title="Nowhere differentiable function">nowhere differentiable</a>. It can be considered as a continuous version of the simple random walk.<a href="#cite_note-Applebaum2004page1337-50">[49]</a><a href="#cite_note-MörtersPeres2010page1-106">[105]</a> The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,<a href="#cite_note-KaratzasShreve2014page61-108">[107]</a><a href="#cite_note-Shreve2004page93-109">[108]</a> which is the subject of <a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a> or invariance principle, also known as the functional central limit theorem.<a href="#cite_note-Kallenberg2002page225and260-110">[109]</a><a href="#cite_note-KaratzasShreve2014page70-111">[110]</a><a href="#cite_note-MörtersPeres2010page131-112">[111]</a> The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.<a href="#cite_note-RogersWilliams2000page1-2">[2]</a><a href="#cite_note-Applebaum2004page1337-50">[49]</a> The process also has many applications and is the main stochastic process used in stochastic calculus.<a href="#cite_note-Klebaner2005-113">[112]</a><a href="#cite_note-KaratzasShreve2014page-114">[113]</a> It plays a central role in quantitative finance,<a href="#cite_note-Applebaum2004page1341-115">[114]</a><a href="#cite_note-KarlinTaylor2012page340-116">[115]</a> where it is used, for example, in the Black–Scholes–Merton model.<a href="#cite_note-Klebaner2005page124-117">[116]</a> The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.<a href="#cite_note-Steele2012page29-3">[3]</a><a href="#cite_note-KaratzasShreve2014page47-118">[117]</a><a href="#cite_note-Wiersema2008page2-119">[118]</a> <div class="mw-heading mw-heading3"><h3 id="Poisson_process">Poisson process</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=10" title="Edit section: Poisson process">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson process</a></div> The Poisson process is a stochastic process that has different forms and definitions.<a href="#cite_note-Tijms2003page1-120">[119]</a><a href="#cite_note-DaleyVere-Jones2006chap2-121">[120]</a> It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.<a href="#cite_note-Tijms2003page1-120">[119]</a> If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.<a href="#cite_note-Tijms2003page1-120">[119]</a><a href="#cite_note-PinskyKarlin2011-122">[121]</a> The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes.<a href="#cite_note-Applebaum2004page1337-50">[49]</a> The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.<a href="#cite_note-Kingman1992page38-123">[122]</a><a href="#cite_note-DaleyVere-Jones2006page19-124">[123]</a> If the parameter constant of the Poisson process is replaced with some non-negative integrable function of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.<a href="#cite_note-Kingman1992page22-125">[124]</a> Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.<a href="#cite_note-KarlinTaylor2012page118-126">[125]</a><a href="#cite_note-Kleinrock1976page61-127">[126]</a> Defined on the real line, the Poisson process can be interpreted as a stochastic process,<a href="#cite_note-Applebaum2004page1337-50">[49]</a><a href="#cite_note-Rosenblatt1962page94-128">[127]</a> among other random objects.<a href="#cite_note-Haenggi2013page10and18-129">[128]</a><a href="#cite_note-ChiuStoyan2013page41and108-130">[129]</a> But then it can be defined on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space or other mathematical spaces,<a href="#cite_note-Kingman1992page11-131">[130]</a> where it is often interpreted as a random set or a random counting measure, instead of a stochastic process.<a href="#cite_note-Haenggi2013page10and18-129">[128]</a><a href="#cite_note-ChiuStoyan2013page41and108-130">[129]</a> In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons.<a href="#cite_note-Stirzaker2000-23">[22]</a><a href="#cite_note-Streit2010page1-132">[131]</a> But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.<a href="#cite_note-Streit2010page1-132">[131]</a><a href="#cite_note-Kingman1992pagev-133">[132]</a> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=11" title="Edit section: Definitions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Stochastic_process">Stochastic process</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=12" title="Edit section: Stochastic process">edit</a>]</div> A stochastic process is defined as a collection of random variables defined on a common <a href="/wiki/Probability_space" title="Probability space">probability space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> is a <a href="/wiki/Sample_space" title="Sample space">sample space</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">-<a href="/wiki/Sigma-algebra" class="mw-redirect" title="Sigma-algebra">algebra</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is a <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a>; and the random variables, indexed by some set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">, all take values in the same mathematical space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, which must be <a href="/wiki/Measurable" class="mw-redirect" title="Measurable">measurable</a> with respect to some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }">.<a href="#cite_note-Lamperti1977page1-29">[28]</a> In other words, for a given probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> and a measurable space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S,\Sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a345c3f30d6b3ee975e3afb6fa610cd3f67494e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.021ex; height:2.843ex;" alt="{\displaystyle (S,\Sigma )}">, a stochastic process is a collection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">-valued random variables, which can be written as:<a href="#cite_note-Florescu2014page293-81">[80]</a> <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X(t):t\in T\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>∈</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X(t):t\in T\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0cc3a859a9ffecd77f44b5540c43d6ecbe5666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.854ex; height:2.843ex;" alt="{\displaystyle \{X(t):t\in T\}.}"></div> Historically, in many problems from the natural sciences a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"> had the meaning of time, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfeb6663c0a903f587cd6d776c387370fc5c4ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.629ex; height:2.843ex;" alt="{\displaystyle X(t)}"> is a random variable representing a value observed at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">.<a href="#cite_note-Borovkov2013page528-134">[133]</a> A stochastic process can also be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X(t,\omega ):t\in T\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>∈</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X(t,\omega ):t\in T\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d3ce53efaf0eaf3a7e67d5d5e679c6da520db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.687ex; height:2.843ex;" alt="{\displaystyle \{X(t,\omega ):t\in T\}}"> to reflect that it is actually a function of two variables, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.965ex; height:2.176ex;" alt="{\displaystyle \omega \in \Omega }">.<a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-LindgrenRootzen2013page11-135">[134]</a> There are other ways to consider a stochastic process, with the above definition being considered the traditional one.<a href="#cite_note-RogersWilliams2000page121-69">[68]</a><a href="#cite_note-Asmussen2003page408-70">[69]</a> For example, a stochastic process can be interpreted or defined as a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c96ff2287e146f1ccab6ae915d9bba2643f7fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.911ex; height:2.676ex;" alt="{\displaystyle S^{T}}">-valued random variable, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c96ff2287e146f1ccab6ae915d9bba2643f7fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.911ex; height:2.676ex;" alt="{\displaystyle S^{T}}"> is the space of all the possible <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> from the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> into the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">.<a href="#cite_note-Kallenberg2002page24-28">[27]</a><a href="#cite_note-RogersWilliams2000page121-69">[68]</a> However this alternative definition as a "function-valued random variable" in general requires additional regularity assumptions to be well-defined.<a href="#cite_note-aumann-136">[135]</a> <div class="mw-heading mw-heading3"><h3 id="Index_set">Index set</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=13" title="Edit section: Index set">edit</a>]</div> The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is called the index set<a href="#cite_note-Parzen1999-4">[4]</a><a href="#cite_note-Florescu2014page294-52">[51]</a> or parameter set<a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-Skorokhod2005page93-137">[136]</a> of the stochastic process. Often this set is some subset of the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>, such as the <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a> or an interval, giving the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> the interpretation of time.<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a> In addition to these sets, the index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> can be another set with a <a href="/wiki/Total_order" title="Total order">total order</a> or a more general set,<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a><a href="#cite_note-Billingsley2008page482-55">[54]</a> such as the Cartesian plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space, where an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"> can represent a point in space.<a href="#cite_note-KarlinTaylor2012page27-49">[48]</a><a href="#cite_note-138">[137]</a> That said, many results and theorems are only possible for stochastic processes with a totally ordered index set.<a href="#cite_note-Skorokhod2005page104-139">[138]</a> <div class="mw-heading mw-heading3"><h3 id="State_space">State space</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=14" title="Edit section: State space">edit</a>]</div> The <a href="/wiki/Mathematical_space" class="mw-redirect" title="Mathematical space">mathematical space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> of a stochastic process is called its state space. This mathematical space can be defined using <a href="/wiki/Integer" title="Integer">integers</a>, <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real lines</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a>, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.<a href="#cite_note-doob1953stochasticP46to47-1">[1]</a><a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-Florescu2014page294-52">[51]</a><a href="#cite_note-Brémaud2014page120-57">[56]</a> <div class="mw-heading mw-heading3"><h3 id="Sample_function">Sample function</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=15" title="Edit section: Sample function">edit</a>]</div> A sample function is a single <a href="/wiki/Outcome_(probability)" title="Outcome (probability)">outcome</a> of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.<a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-Florescu2014page296-140">[139]</a> More precisely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X(t,\omega ):t\in T\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>∈</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X(t,\omega ):t\in T\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d3ce53efaf0eaf3a7e67d5d5e679c6da520db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.687ex; height:2.843ex;" alt="{\displaystyle \{X(t,\omega ):t\in T\}}"> is a stochastic process, then for any point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.965ex; height:2.176ex;" alt="{\displaystyle \omega \in \Omega }">, the <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a> <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(\cdot ,\omega ):T\rightarrow S,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>T</mi> <mo stretchy="false">→</mo> <mi>S</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(\cdot ,\omega ):T\rightarrow S,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2718b5d432f366d7699c877cc704a50c3ee1db63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.249ex; height:2.843ex;" alt="{\displaystyle X(\cdot ,\omega ):T\rightarrow S,}"></div> is called a sample function, a realization, or, particularly when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is interpreted as time, a sample path of the stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X(t,\omega ):t\in T\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>∈</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X(t,\omega ):t\in T\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12d3ce53efaf0eaf3a7e67d5d5e679c6da520db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.687ex; height:2.843ex;" alt="{\displaystyle \{X(t,\omega ):t\in T\}}">.<a href="#cite_note-RogersWilliams2000page121b-51">[50]</a> This means that for a fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \in \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.965ex; height:2.176ex;" alt="{\displaystyle \omega \in \Omega }">, there exists a sample function that maps the index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> to the state space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">.<a href="#cite_note-Lamperti1977page1-29">[28]</a> Other names for a sample function of a stochastic process include trajectory, path function<a href="#cite_note-Billingsley2008page493-141">[140]</a> or path.<a href="#cite_note-Øksendal2003page10-142">[141]</a> <div class="mw-heading mw-heading3"><h3 id="Increment">Increment</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=16" title="Edit section: Increment">edit</a>]</div> An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X(t):t\in T\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>∈</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X(t):t\in T\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7217d98e276b7c261071ee4e862c8630ea616ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.208ex; height:2.843ex;" alt="{\displaystyle \{X(t):t\in T\}}"> is a stochastic process with state space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> and index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=[0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=[0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d5792c68149c5c2f3e434e8c119a702ced9846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.806ex; height:2.843ex;" alt="{\displaystyle T=[0,\infty )}">, then for any two non-negative numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}\in [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}\in [0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06494357e499250219583fb45ba3459986d12806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.806ex; height:2.843ex;" alt="{\displaystyle t_{1}\in [0,\infty )}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{2}\in [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{2}\in [0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8503b7393f08fb2d5f14fc1d022d918a1b8c7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.806ex; height:2.843ex;" alt="{\displaystyle t_{2}\in [0,\infty )}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}\leq t_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1}\leq t_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64ae8eb3430c6f0d75d0d8f09c1e0ed8af4cb92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.886ex; height:2.343ex;" alt="{\displaystyle t_{1}\leq t_{2}}">, the difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t_{2}}-X_{t_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t_{2}}-X_{t_{1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c007b9a100a5666b613bcb15f4f70d52e44b2a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.004ex; height:2.843ex;" alt="{\displaystyle X_{t_{2}}-X_{t_{1}}}"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">-valued random variable known as an increment.<a href="#cite_note-KarlinTaylor2012page27-49">[48]</a><a href="#cite_note-Applebaum2004page1337-50">[49]</a> When interested in the increments, often the state space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is the real line or the natural numbers, but it can be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space or more abstract spaces such as <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>.<a href="#cite_note-Applebaum2004page1337-50">[49]</a> <div class="mw-heading mw-heading3"><h3 id="Further_definitions">Further definitions</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=17" title="Edit section: Further definitions">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Law">Law</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=18" title="Edit section: Law">edit</a>]</div> For a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \rightarrow S^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \rightarrow S^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/087447398e813d81b4e4612fd7623390578a108b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.217ex; height:2.676ex;" alt="{\displaystyle X\colon \Omega \rightarrow S^{T}}"> defined on the probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}">, the law of stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is defined as the <a href="/wiki/Pushforward_measure" title="Pushforward measure">image measure</a>: <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =P\circ X^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>P</mi> <mo>∘</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =P\circ X^{-1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aaa4c4dfdcc71e386d78cddc468cb979189bf3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.417ex; height:3.176ex;" alt="{\displaystyle \mu =P\circ X^{-1},}"></div> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is a probability measure, the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }"> denotes function composition and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71f75a18781a402f9fb09972f99e092bf21aa877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.33ex; height:2.676ex;" alt="{\displaystyle X^{-1}}"> is the pre-image of the measurable function or, equivalently, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c96ff2287e146f1ccab6ae915d9bba2643f7fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.911ex; height:2.676ex;" alt="{\displaystyle S^{T}}">-valued random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c96ff2287e146f1ccab6ae915d9bba2643f7fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.911ex; height:2.676ex;" alt="{\displaystyle S^{T}}"> is the space of all the possible <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">-valued functions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}">, so the law of a stochastic process is a probability measure.<a href="#cite_note-Kallenberg2002page24-28">[27]</a><a href="#cite_note-RogersWilliams2000page121-69">[68]</a><a href="#cite_note-FrizVictoir2010page571-143">[142]</a><a href="#cite_note-Resnick2013page40-144">[143]</a> For a measurable subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c96ff2287e146f1ccab6ae915d9bba2643f7fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.911ex; height:2.676ex;" alt="{\displaystyle S^{T}}">, the pre-image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> gives <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{-1}(B)=\{\omega \in \Omega :X(\omega )\in B\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{-1}(B)=\{\omega \in \Omega :X(\omega )\in B\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/613f6fbcab13d949e540fa194fa227ebb90a856c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.715ex; height:3.176ex;" alt="{\displaystyle X^{-1}(B)=\{\omega \in \Omega :X(\omega )\in B\},}"></div> so the law of a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> can be written as:<a href="#cite_note-Lamperti1977page1-29">[28]</a> <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (B)=P(\{\omega \in \Omega :X(\omega )\in B\}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (B)=P(\{\omega \in \Omega :X(\omega )\in B\}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221e1244c4a1dd6eb832db40c4d7fae1fb4a4b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.341ex; height:2.843ex;" alt="{\displaystyle \mu (B)=P(\{\omega \in \Omega :X(\omega )\in B\}).}"></div> The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution.<a href="#cite_note-Borovkov2013page528-134">[133]</a><a href="#cite_note-FrizVictoir2010page571-143">[142]</a><a href="#cite_note-Whitt2006page23-145">[144]</a><a href="#cite_note-ApplebaumBook2004page4-146">[145]</a><a href="#cite_note-RevuzYor2013page10-147">[146]</a> <div class="mw-heading mw-heading4"><h4 id="Finite-dimensional_probability_distributions">Finite-dimensional probability distributions</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=19" title="Edit section: Finite-dimensional probability distributions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite-dimensional_distribution" title="Finite-dimensional distribution">Finite-dimensional distribution</a></div> For a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> with law <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }">, its finite-dimensional distribution for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},\dots ,t_{n}\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1},\dots ,t_{n}\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a87f94ed161df0e7f64afdb1e06872490967bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.607ex; height:2.509ex;" alt="{\displaystyle t_{1},\dots ,t_{n}\in T}"> is defined as: <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{t_{1},\dots ,t_{n}}=P\circ (X({t_{1}}),\dots ,X({t_{n}}))^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mi>P</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{t_{1},\dots ,t_{n}}=P\circ (X({t_{1}}),\dots ,X({t_{n}}))^{-1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5824dfc4529fdcf79a0a20fe0828638fb7090cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.994ex; height:3.343ex;" alt="{\displaystyle \mu _{t_{1},\dots ,t_{n}}=P\circ (X({t_{1}}),\dots ,X({t_{n}}))^{-1},}"></div> This measure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{t_{1},..,t_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{t_{1},..,t_{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524b5229b797357e587ae012be097d9cbbb703a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.447ex; height:2.343ex;" alt="{\displaystyle \mu _{t_{1},..,t_{n}}}"> is the joint distribution of the random vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X({t_{1}}),\dots ,X({t_{n}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X({t_{1}}),\dots ,X({t_{n}}))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4abce4e2541966d9c824830c5246cf00e2e5dd0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.518ex; height:2.843ex;" alt="{\displaystyle (X({t_{1}}),\dots ,X({t_{n}}))}">; it can be viewed as a "projection" of the law <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> onto a finite subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">.<a href="#cite_note-Kallenberg2002page24-28">[27]</a><a href="#cite_note-RogersWilliams2000page123-148">[147]</a> For any measurable subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-fold <a href="/wiki/Cartesian_power" class="mw-redirect" title="Cartesian power">Cartesian power</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{n}=S\times \dots \times S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>S</mi> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{n}=S\times \dots \times S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9743c7006613bd800da45a69ce249c6a40c57ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.241ex; height:2.343ex;" alt="{\displaystyle S^{n}=S\times \dots \times S}">, the finite-dimensional distributions of a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> can be written as:<a href="#cite_note-Lamperti1977page1-29">[28]</a> <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{t_{1},\dots ,t_{n}}(C)=P{\Big (}{\big \{}\omega \in \Omega :{\big (}X_{t_{1}}(\omega ),\dots ,X_{t_{n}}(\omega ){\big )}\in C{\big \}}{\Big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mi>ω</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>∈</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{t_{1},\dots ,t_{n}}(C)=P{\Big (}{\big \{}\omega \in \Omega :{\big (}X_{t_{1}}(\omega ),\dots ,X_{t_{n}}(\omega ){\big )}\in C{\big \}}{\Big )}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d4be29f417d006975f5be6824c2feb6e3a3e344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.634ex; height:4.843ex;" alt="{\displaystyle \mu _{t_{1},\dots ,t_{n}}(C)=P{\Big (}{\big \{}\omega \in \Omega :{\big (}X_{t_{1}}(\omega ),\dots ,X_{t_{n}}(\omega ){\big )}\in C{\big \}}{\Big )}.}"></div> The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.<a href="#cite_note-Rosenthal2006page177-58">[57]</a> <div class="mw-heading mw-heading4"><h4 id="Stationarity">Stationarity</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=20" title="Edit section: Stationarity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stationary_process" title="Stationary process">Stationary process</a></div> Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a stationary stochastic process, then for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"> the random variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}"> has the same distribution, which means that for any set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> index set values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},\dots ,t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1},\dots ,t_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34a1feec735ff72e72b1b6e15b45a3cef67cade3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.13ex; height:2.343ex;" alt="{\displaystyle t_{1},\dots ,t_{n}}">, the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> random variables <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t_{1}},\dots X_{t_{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t_{1}},\dots X_{t_{n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1fc37d32d5937697f3f2239df61ad039fbc3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.088ex; height:2.843ex;" alt="{\displaystyle X_{t_{1}},\dots X_{t_{n}},}"></div> all have the same <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a>. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.<a href="#cite_note-Lamperti1977page6-149">[148]</a><a href="#cite_note-GikhmanSkorokhod1969page4-150">[149]</a> But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.<a href="#cite_note-Lamperti1977page6-149">[148]</a><a href="#cite_note-Adler2010page14-151">[150]</a><a href="#cite_note-ChiuStoyan2013page112-152">[151]</a> When the index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations.<a href="#cite_note-Lamperti1977page6-149">[148]</a> The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same.<a href="#cite_note-Doob1990page94-153">[152]</a> A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.<a href="#cite_note-Lamperti1977page6-149">[148]</a> A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or continuous-time stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is said to be stationary in the wide sense, then the process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> has a finite second moment for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"> and the covariance of the two random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t+h}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>+</mo> <mi>h</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t+h}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd42440349572614fdd9732f222a5967966b488e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.976ex; height:2.509ex;" alt="{\displaystyle X_{t+h}}"> depends only on the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}">.<a href="#cite_note-Doob1990page94-153">[152]</a><a href="#cite_note-Florescu2014page298-154">[153]</a> <a href="/wiki/Aleksandr_Khinchin" title="Aleksandr Khinchin">Khinchin</a> introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.<a href="#cite_note-Florescu2014page298-154">[153]</a><a href="#cite_note-GikhmanSkorokhod1969page8-155">[154]</a> <div class="mw-heading mw-heading4"><h4 id="Filtration">Filtration</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=21" title="Edit section: Filtration">edit</a>]</div> A <a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">filtration</a> is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some <a href="/wiki/Total_order" title="Total order">total order</a> relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\mathcal {F}}_{t}\}_{t\in T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\mathcal {F}}_{t}\}_{t\in T}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502d06196a101b3d5fc05c9d6af22e88e8138a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.901ex; height:2.843ex;" alt="{\displaystyle \{{\mathcal {F}}_{t}\}_{t\in T}}">, on a probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> is a family of sigma-algebras such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{s}\subseteq {\mathcal {F}}_{t}\subseteq {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⊆</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{s}\subseteq {\mathcal {F}}_{t}\subseteq {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db36797ea37ac95fa216064ec73cf20727fbfca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.295ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{s}\subseteq {\mathcal {F}}_{t}\subseteq {\mathcal {F}}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\leq t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≤</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\leq t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c12f663b9eb8759c0da898cc542d7cb3bcf5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.029ex; height:2.176ex;" alt="{\displaystyle s\leq t}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t,s\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>,</mo> <mi>s</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,s\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0955558a4194833987ad85e254a6466f3a67a0c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.441ex; height:2.509ex;" alt="{\displaystyle t,s\in T}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≤</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leq }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440568a09c3bfdf0e1278bfa79eb137c04e94035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }"> denotes the total order of the index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">.<a href="#cite_note-Florescu2014page294-52">[51]</a> With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}">, which can be interpreted as time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">.<a href="#cite_note-Florescu2014page294-52">[51]</a><a href="#cite_note-Williams1991page93-156">[155]</a> The intuition behind a filtration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.497ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{t}}"> is that as time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> passes, more and more information on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}"> is known or available, which is captured in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{t}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.497ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{t}}">, resulting in finer and finer partitions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }">.<a href="#cite_note-Klebaner2005page22-157">[156]</a><a href="#cite_note-MörtersPeres2010page37-158">[157]</a> <div class="mw-heading mw-heading4"><h4 id="Modification">Modification</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=22" title="Edit section: Modification">edit</a>]</div> A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> that has the same index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">, state space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">, and probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\cal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\cal {F}},P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52145a67f2c90f7a3b14c5bb603cd6d74ea247c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\cal {F}},P)}"> as another stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> is said to be a modification of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"> the following <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X_{t}=Y_{t})=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X_{t}=Y_{t})=1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597b85d90932a65773d5521ed76f017c4dcf8b2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.488ex; height:2.843ex;" alt="{\displaystyle P(X_{t}=Y_{t})=1,}"></div> holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law<a href="#cite_note-RogersWilliams2000page130-159">[158]</a> and they are said to be stochastically equivalent or equivalent.<a href="#cite_note-Borovkov2013page530-160">[159]</a> Instead of modification, the term version is also used,<a href="#cite_note-Adler2010page14-151">[150]</a><a href="#cite_note-Klebaner2005page48-161">[160]</a><a href="#cite_note-Øksendal2003page14-162">[161]</a><a href="#cite_note-Florescu2014page472-163">[162]</a> however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.<a href="#cite_note-RevuzYor2013page18-164">[163]</a><a href="#cite_note-FrizVictoir2010page571-143">[142]</a> If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the <a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a> says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.<a href="#cite_note-Øksendal2003page14-162">[161]</a><a href="#cite_note-Florescu2014page472-163">[162]</a><a href="#cite_note-ApplebaumBook2004page20-165">[164]</a> The theorem can also be generalized to random fields so the index set is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space<a href="#cite_note-Kunita1997page31-166">[165]</a> as well as to stochastic processes with <a href="/wiki/Metric_spaces" class="mw-redirect" title="Metric spaces">metric spaces</a> as their state spaces.<a href="#cite_note-Kallenberg2002page-167">[166]</a> <div class="mw-heading mw-heading4"><h4 id="Indistinguishable">Indistinguishable</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=23" title="Edit section: Indistinguishable">edit</a>]</div> Two stochastic processes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> defined on the same probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> with the same index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> and set space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> are said be indistinguishable if the following <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X_{t}=Y_{t}{\text{ for all }}t\in T)=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>t</mi> <mo>∈</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X_{t}=Y_{t}{\text{ for all }}t\in T)=1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c12532d934c0e9f7ccc82f1a290edb209d447c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.788ex; height:2.843ex;" alt="{\displaystyle P(X_{t}=Y_{t}{\text{ for all }}t\in T)=1,}"></div> holds.<a href="#cite_note-FrizVictoir2010page571-143">[142]</a><a href="#cite_note-RogersWilliams2000page130-159">[158]</a> If two <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> are modifications of each other and are <a href="/wiki/Almost_surely_continuous" class="mw-redirect" title="Almost surely continuous">almost surely continuous</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> are indistinguishable.<a href="#cite_note-JeanblancYor2009page11-168">[167]</a> <div class="mw-heading mw-heading4"><h4 id="Separability">Separability</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=24" title="Edit section: Separability">edit</a>]</div> Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a <a href="/wiki/Separable_space" title="Separable space">separable space</a>,<a href="#cite_note-169">[b]</a> which means that the index set has a dense countable subset.<a href="#cite_note-Adler2010page14-151">[150]</a><a href="#cite_note-Ito2006page32-170">[168]</a> More precisely, a real-valued continuous-time stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> with a probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\cal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\cal {F}},P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52145a67f2c90f7a3b14c5bb603cd6d74ea247c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\cal {F}},P)}"> is separable if its index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> has a dense countable subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊂</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subset T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3968c1e805d40ed499802acacfacca97dfabd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.517ex; height:2.176ex;" alt="{\displaystyle U\subset T}"> and there is a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{0}\subset \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊂</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{0}\subset \Omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96f01a9a38d481400f585fffc08f8d6601f2ac59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.509ex; height:2.509ex;" alt="{\displaystyle \Omega _{0}\subset \Omega }"> of probability zero, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\Omega _{0})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\Omega _{0})=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a246b15f0656ed2fed4fc327a3502046f6b4be0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.548ex; height:2.843ex;" alt="{\displaystyle P(\Omega _{0})=0}">, such that for every open set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\subset T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>⊂</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\subset T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd19cc42d8c42e2e557137014251083ba689b456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.561ex; height:2.176ex;" alt="{\displaystyle G\subset T}"> and every closed set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\subset \textstyle R=(-\infty ,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊂</mo> <mstyle displaystyle="false" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subset \textstyle R=(-\infty ,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db918c3b861f8d7be9610d9d6423486888f3195" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.001ex; height:2.843ex;" alt="{\displaystyle F\subset \textstyle R=(-\infty ,\infty )}">, the two events <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\cap U\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∈</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>t</mi> <mo>∈</mo> <mi>G</mi> <mo>∩</mo> <mi>U</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\cap U\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5b9705ad73bded760311dc87bbbe97dbcd49e70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.513ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\cap U\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∈</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>t</mi> <mo>∈</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc08b8b4b583ef98a4ad09f2f1291c85722ddf02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.148ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\}}"> differ from each other at most on a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d09f58739bf722ad06801001704ea92360f2b96c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \Omega _{0}}">.<a href="#cite_note-GikhmanSkorokhod1969page150-171">[169]</a><a href="#cite_note-Todorovic2012page19-172">[170]</a><a href="#cite_note-Molchanov2005page340-173">[171]</a> The definition of separability<a href="#cite_note-176">[c]</a> can also be stated for other index sets and state spaces,<a href="#cite_note-GusakKukush2010page22-177">[174]</a> such as in the case of random fields, where the index set as well as the state space can be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space.<a href="#cite_note-AdlerTaylor2009page7-31">[30]</a><a href="#cite_note-Adler2010page14-151">[150]</a> The concept of separability of a stochastic process was introduced by <a href="/wiki/Joseph_Doob" class="mw-redirect" title="Joseph Doob">Joseph Doob</a>,.<a href="#cite_note-Ito2006page32-170">[168]</a> The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.<a href="#cite_note-Billingsley2008page526-174">[172]</a> Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable.<a href="#cite_note-Doob1990page56-178">[175]</a> A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.<a href="#cite_note-Ito2006page32-170">[168]</a><a href="#cite_note-Todorovic2012page19-172">[170]</a><a href="#cite_note-Khoshnevisan2006page155-179">[176]</a> Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.<a href="#cite_note-Skorokhod2005page93-137">[136]</a> <div class="mw-heading mw-heading4"><h4 id="Independence">Independence</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=25" title="Edit section: Independence">edit</a>]</div> Two stochastic processes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> defined on the same probability space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d77104a5c3c49cc0634dcf6908db7ad45f738d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}"> with the same index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> are said be independent if for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }"> and for every choice of epochs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},\ldots ,t_{n}\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1},\ldots ,t_{n}\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f716f5b0a4ac4c6438cd38f258beb4c0cbbd4f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.607ex; height:2.509ex;" alt="{\displaystyle t_{1},\ldots ,t_{n}\in T}">, the random vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(X(t_{1}),\ldots ,X(t_{n})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(X(t_{1}),\ldots ,X(t_{n})\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95bd90485ef63f13e126bcd90da7eea74c97a13b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.518ex; height:2.843ex;" alt="{\displaystyle \left(X(t_{1}),\ldots ,X(t_{n})\right)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(Y(t_{1}),\ldots ,Y(t_{n})\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(Y(t_{1}),\ldots ,Y(t_{n})\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66931d08caf85307b063d4a59e45da0a3354ac71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.105ex; height:2.843ex;" alt="{\displaystyle \left(Y(t_{1}),\ldots ,Y(t_{n})\right)}"> are independent.<a href="#cite_note-Lapidoth-180">[177]</a>: p. 515  <div class="mw-heading mw-heading4"><h4 id="Uncorrelatedness">Uncorrelatedness</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=26" title="Edit section: Uncorrelatedness">edit</a>]</div> Two stochastic processes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{t}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{t}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d06db272f40cad57e421e3a38af88597d0709a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.075ex; height:2.843ex;" alt="{\displaystyle \left\{X_{t}\right\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{Y_{t}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{Y_{t}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f6c45b3e23284b76bf9081ac85c1d518ea6fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.501ex; height:2.843ex;" alt="{\displaystyle \left\{Y_{t}\right\}}"> are called uncorrelated if their cross-covariance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} \left[\left(X(t_{1})-\mu _{X}(t_{1})\right)\left(Y(t_{2})-\mu _{Y}(t_{2})\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} \left[\left(X(t_{1})-\mu _{X}(t_{1})\right)\left(Y(t_{2})-\mu _{Y}(t_{2})\right)\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf123985d84cdc25587546891caea6d23dc4f6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.677ex; height:2.843ex;" alt="{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} \left[\left(X(t_{1})-\mu _{X}(t_{1})\right)\left(Y(t_{2})-\mu _{Y}(t_{2})\right)\right]}"> is zero for all times.<a href="#cite_note-KunIlPark-181">[178]</a>: p. 142  Formally: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ uncorrelated}}\quad \iff \quad \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> uncorrelated</mtext> </mrow> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi mathvariant="normal">K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ uncorrelated}}\quad \iff \quad \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18e141616473b88be7ac4d7ca1f26ec9e9a87c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.407ex; height:2.843ex;" alt="{\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ uncorrelated}}\quad \iff \quad \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}}">.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Independence_implies_uncorrelatedness">Independence implies uncorrelatedness</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=27" title="Edit section: Independence implies uncorrelatedness">edit</a>]</div> If two stochastic processes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"> are independent, then they are also uncorrelated.<a href="#cite_note-KunIlPark-181">[178]</a>: p. 151  <div class="mw-heading mw-heading4"><h4 id="Orthogonality">Orthogonality</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=28" title="Edit section: Orthogonality">edit</a>]</div> Two stochastic processes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{t}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{t}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d06db272f40cad57e421e3a38af88597d0709a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.075ex; height:2.843ex;" alt="{\displaystyle \left\{X_{t}\right\}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{Y_{t}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{Y_{t}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f6c45b3e23284b76bf9081ac85c1d518ea6fca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.501ex; height:2.843ex;" alt="{\displaystyle \left\{Y_{t}\right\}}"> are called orthogonal if their cross-correlation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} [X(t_{1}){\overline {Y(t_{2})}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} [X(t_{1}){\overline {Y(t_{2})}}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93d58e51ab83745d727396e86c894660ee44857e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.679ex; height:3.676ex;" alt="{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} [X(t_{1}){\overline {Y(t_{2})}}]}"> is zero for all times.<a href="#cite_note-KunIlPark-181">[178]</a>: p. 142  Formally: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ orthogonal}}\quad \iff \quad \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> orthogonal</mtext> </mrow> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <msub> <mi mathvariant="normal">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Y</mi> </mrow> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ orthogonal}}\quad \iff \quad \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af0fa951e2964129d5561a57172895d54d95bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.496ex; height:2.843ex;" alt="{\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ orthogonal}}\quad \iff \quad \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}}">.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Skorokhod_space">Skorokhod space</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=29" title="Edit section: Skorokhod space">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></div> A Skorokhod space, also written as Skorohod space, is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}">, and take values on the real line or on some metric space.<a href="#cite_note-Whitt2006page78-182">[179]</a><a href="#cite_note-GusakKukush2010page24-183">[180]</a><a href="#cite_note-Bogachev2007Vol2page53-184">[181]</a> Such functions are known as càdlàg or cadlag functions, based on the acronym of the French phrase continue à droite, limite à gauche.<a href="#cite_note-Whitt2006page78-182">[179]</a><a href="#cite_note-Klebaner2005page4-185">[182]</a> A Skorokhod function space, introduced by <a href="/wiki/Anatoliy_Skorokhod" title="Anatoliy Skorokhod">Anatoliy Skorokhod</a>,<a href="#cite_note-Bogachev2007Vol2page53-184">[181]</a> is often denoted with the letter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}">,<a href="#cite_note-Whitt2006page78-182">[179]</a><a href="#cite_note-GusakKukush2010page24-183">[180]</a><a href="#cite_note-Bogachev2007Vol2page53-184">[181]</a><a href="#cite_note-Klebaner2005page4-185">[182]</a> so the function space is also referred to as space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}">.<a href="#cite_note-Whitt2006page78-182">[179]</a><a href="#cite_note-Asmussen2003page420-186">[183]</a><a href="#cite_note-Billingsley2013page121-187">[184]</a> The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D[0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D[0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d18229131d1ad9cd7cff586ad63243aa6acaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.577ex; height:2.843ex;" alt="{\displaystyle D[0,1]}"> denotes the space of càdlàg functions defined on the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}">.<a href="#cite_note-Klebaner2005page4-185">[182]</a><a href="#cite_note-Billingsley2013page121-187">[184]</a><a href="#cite_note-Bass2011page34-188">[185]</a> Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space.<a href="#cite_note-Bogachev2007Vol2page53-184">[181]</a><a href="#cite_note-Asmussen2003page420-186">[183]</a> Such spaces contain continuous functions, which correspond to sample functions of the Wiener process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space.<a href="#cite_note-Billingsley2013page121-187">[184]</a><a href="#cite_note-BinghamKiesel2013page154-189">[186]</a> <div class="mw-heading mw-heading4"><h4 id="Regularity">Regularity</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=30" title="Edit section: Regularity">edit</a>]</div> In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues.<a href="#cite_note-Borovkov2013page532-190">[187]</a><a href="#cite_note-Khoshnevisan2006page148to165-191">[188]</a> For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.<a href="#cite_note-Todorovic2012page22-192">[189]</a><a href="#cite_note-Whitt2006page79-193">[190]</a> <div class="mw-heading mw-heading2"><h2 id="Further_examples">Further examples</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=31" title="Edit section: Further examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Markov_processes_and_chains">Markov processes and chains</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=32" title="Edit section: Markov processes and chains">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></div> Markov processes are stochastic processes, traditionally in <a href="/wiki/Discrete_time_and_continuous_time" title="Discrete time and continuous time">discrete or continuous time</a>, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.<a href="#cite_note-Serfozo2009page2-194">[191]</a><a href="#cite_note-Rozanov2012page58-195">[192]</a> The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes<a href="#cite_note-Ross1996page235and358-196">[193]</a> in continuous time, while <a href="/wiki/Random_walk" title="Random walk">random walks</a> on the integers and the <a href="/wiki/Gambler%27s_ruin" title="Gambler's ruin">gambler's ruin</a> problem are examples of Markov processes in discrete time.<a href="#cite_note-Florescu2014page373-197">[194]</a><a href="#cite_note-KarlinTaylor2012page49-198">[195]</a> A Markov chain is a type of Markov process that has either discrete <a href="/wiki/State_space" class="mw-redirect" title="State space">state space</a> or discrete index set (often representing time), but the precise definition of a Markov chain varies.<a href="#cite_note-Asmussen2003page7-199">[196]</a> For example, it is common to define a Markov chain as a Markov process in either <a href="/wiki/Continuous_and_discrete_variables" class="mw-redirect" title="Continuous and discrete variables">discrete or continuous time</a> with a countable state space (thus regardless of the nature of time),<a href="#cite_note-Parzen1999page188-200">[197]</a><a href="#cite_note-KarlinTaylor2012page29-201">[198]</a><a href="#cite_note-Lamperti1977chap6-202">[199]</a><a href="#cite_note-Ross1996page174and231-203">[200]</a> but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).<a href="#cite_note-Asmussen2003page7-199">[196]</a> It has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used by researchers like <a href="/wiki/Joseph_Doob" class="mw-redirect" title="Joseph Doob">Joseph Doob</a> and <a href="/wiki/Kai_Lai_Chung" class="mw-redirect" title="Kai Lai Chung">Kai Lai Chung</a>.<a href="#cite_note-MeynTweedie2009-204">[201]</a> Markov processes form an important class of stochastic processes and have applications in many areas.<a href="#cite_note-LatoucheRamaswami1999-40">[39]</a><a href="#cite_note-KarlinTaylor2012page47-205">[202]</a> For example, they are the basis for a general stochastic simulation method known as <a href="/wiki/Markov_chain_Monte_Carlo" title="Markov chain Monte Carlo">Markov chain Monte Carlo</a>, which is used for simulating random objects with specific probability distributions, and has found application in <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>.<a href="#cite_note-RubinsteinKroese2011page225-206">[203]</a><a href="#cite_note-GamermanLopes2006-207">[204]</a> The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.<a href="#cite_note-Rozanov2012page61-208">[205]</a><a href="#cite_note-209">[206]</a><a href="#cite_note-Bremaud2013page253-210">[207]</a> <div class="mw-heading mw-heading3"><h3 id="Martingale">Martingale</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=33" title="Edit section: Martingale">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale (probability theory)</a></div> A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,<a href="#cite_note-Klebaner2005page65-211">[208]</a><a href="#cite_note-KaratzasShreve2014page11-212">[209]</a><a href="#cite_note-Williams1991page93-156">[155]</a> but they can also be complex-valued<a href="#cite_note-Doob1990page292-213">[210]</a> or even more general.<a href="#cite_note-Pisier2016-214">[211]</a> A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time.<a href="#cite_note-Klebaner2005page65-211">[208]</a><a href="#cite_note-KaratzasShreve2014page11-212">[209]</a> For a <a href="/wiki/Sequence" title="Sequence">sequence</a> of <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},X_{3},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},X_{3},\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7560e97305db33eaed1fc9835fe1487808a9dbc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.761ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{2},X_{3},\dots }"> with zero mean, the stochastic process formed from the successive partial sums <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{1}+X_{2},X_{1}+X_{2}+X_{3},\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{1},X_{1}+X_{2},X_{1}+X_{2}+X_{3},\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5281324153001c691b88e6e2e22782a80bf8d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.217ex; height:2.509ex;" alt="{\displaystyle X_{1},X_{1}+X_{2},X_{1}+X_{2}+X_{3},\dots }"> is a discrete-time martingale.<a href="#cite_note-Steele2012page12-215">[212]</a> In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.<a href="#cite_note-HallHeyde2014page2-216">[213]</a> Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process.<a href="#cite_note-KaratzasShreve2014page11-212">[209]</a> Martingales can also be built from other martingales.<a href="#cite_note-Steele2012page12-215">[212]</a> For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.<a href="#cite_note-Klebaner2005page65-211">[208]</a><a href="#cite_note-Steele2012page115-217">[214]</a> Martingales mathematically formalize the idea of a 'fair game' where it is possible form reasonable expectations for payoffs,<a href="#cite_note-Ross1996page295-218">[215]</a> and they were originally developed to show that it is not possible to gain an 'unfair' advantage in such a game.<a href="#cite_note-Steele2012page11-219">[216]</a> But now they are used in many areas of probability, which is one of the main reasons for studying them.<a href="#cite_note-Williams1991page93-156">[155]</a><a href="#cite_note-Steele2012page11-219">[216]</a><a href="#cite_note-Kallenberg2002page96-220">[217]</a> Many problems in probability have been solved by finding a martingale in the problem and studying it.<a href="#cite_note-Steele2012page371-221">[218]</a> Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to <a href="/wiki/Martingale_convergence_theorem" class="mw-redirect" title="Martingale convergence theorem">martingale convergence theorems</a>.<a href="#cite_note-HallHeyde2014page2-216">[213]</a><a href="#cite_note-Steele2012page22-222">[219]</a><a href="#cite_note-GrimmettStirzaker2001page336-223">[220]</a> Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<a href="#cite_note-GlassermanKou2006-224">[221]</a> They have found applications in areas in probability theory such as queueing theory and Palm calculus<a href="#cite_note-BaccelliBremaud2013-225">[222]</a> and other fields such as economics<a href="#cite_note-HallHeyde2014pageX-226">[223]</a> and finance.<a href="#cite_note-MusielaRutkowski2006-17">[17]</a> <div class="mw-heading mw-heading3"><h3 id="Lévy_process">Lévy process</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=34" title="Edit section: Lévy process">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></div> Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.<a href="#cite_note-Applebaum2004page1337-50">[49]</a><a href="#cite_note-Bertoin1998pageVIII-227">[224]</a> These processes have many applications in fields such as finance, fluid mechanics, physics and biology.<a href="#cite_note-Applebaum2004page1336-228">[225]</a><a href="#cite_note-ApplebaumBook2004page69-229">[226]</a> The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a Lévy process if for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> non-negatives numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq t_{1}\leq \dots \leq t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq t_{1}\leq \dots \leq t_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dde63b2a6a06cf603a6457d0c3f9c9ef8ed46d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.133ex; height:2.509ex;" alt="{\displaystyle 0\leq t_{1}\leq \dots \leq t_{n}}">, the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"> increments <div class="center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t_{2}}-X_{t_{1}},\dots ,X_{t_{n}}-X_{t_{n-1}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t_{2}}-X_{t_{1}},\dots ,X_{t_{n}}-X_{t_{n-1}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd3adc835e84ef7814eff2f4ed1c8c14e4c41bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.805ex; height:2.843ex;" alt="{\displaystyle X_{t_{2}}-X_{t_{1}},\dots ,X_{t_{n}}-X_{t_{n-1}},}"></div> are all independent of each other, and the distribution of each increment only depends on the difference in time.<a href="#cite_note-Applebaum2004page1337-50">[49]</a> A Lévy process can be defined such that its state space is some abstract mathematical space, such as a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>, but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=[0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3abb1be05fb1164e34416ad00ac5d9705bafff58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.342ex; height:2.843ex;" alt="{\displaystyle I=[0,\infty )}">, which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and <a href="/wiki/Subordinator_(mathematics)" title="Subordinator (mathematics)">subordinators</a> are all Lévy processes.<a href="#cite_note-Applebaum2004page1337-50">[49]</a><a href="#cite_note-Bertoin1998pageVIII-227">[224]</a> <div class="mw-heading mw-heading3"><h3 id="Random_field">Random field</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=35" title="Edit section: Random field">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Random_field" title="Random field">Random field</a></div> A random field is a collection of random variables indexed by a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space or some manifold. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line.<a href="#cite_note-AdlerTaylor2009page7-31">[30]</a> But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions.<a href="#cite_note-GikhmanSkorokhod1969page1-5">[5]</a><a href="#cite_note-Lamperti1977page1-29">[28]</a><a href="#cite_note-KoralovSinai2007page171-230">[227]</a> If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.<a href="#cite_note-ApplebaumBook2004page19-231">[228]</a> <div class="mw-heading mw-heading3"><h3 id="Point_process">Point process</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=36" title="Edit section: Point process">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Point_process" title="Point process">Point process</a></div> A point process is a collection of points randomly located on some mathematical space such as the real line, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space, or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field.<a href="#cite_note-ChiuStoyan2013page109-232">[229]</a> There are different interpretations of a point process, such a random counting measure or a random set.<a href="#cite_note-ChiuStoyan2013page108-233">[230]</a><a href="#cite_note-Haenggi2013page10-234">[231]</a> Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process,<a href="#cite_note-DaleyVere-Jones2006page194-235">[232]</a><a href="#cite_note-CoxIsham1980page3-236">[233]</a> though it has been remarked that the difference between point processes and stochastic processes is not clear.<a href="#cite_note-CoxIsham1980page3-236">[233]</a> Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space<a href="#cite_note-239">[d]</a> on which it is defined, such as the real line or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space.<a href="#cite_note-KarlinTaylor2012page31-240">[236]</a><a href="#cite_note-Schmidt2014page99-241">[237]</a> Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.<a href="#cite_note-DaleyVere-Jones200-242">[238]</a><a href="#cite_note-CoxIsham1980page3-236">[233]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=37" title="Edit section: History">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Early_probability_theory">Early probability theory</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=38" title="Edit section: Early probability theory">edit</a>]</div> Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago,<a href="#cite_note-David1955-243">[239]</a> but very little analysis on them was done in terms of probability.<a href="#cite_note-Maistrov2014page1-244">[240]</a> The year 1654 is often considered the birth of probability theory when French mathematicians <a href="/wiki/Pierre_Fermat" class="mw-redirect" title="Pierre Fermat">Pierre Fermat</a> and <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> had a written correspondence on probability, motivated by a <a href="/wiki/Problem_of_points" title="Problem of points">gambling problem</a>.<a href="#cite_note-Seneta2006page1-245">[241]</a><a href="#cite_note-Tabak2014page24to26-246">[242]</a> But there was earlier mathematical work done on the probability of gambling games such as Liber de Ludo Aleae by <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>, written in the 16th century but posthumously published later in 1663.<a href="#cite_note-Bellhouse2005-247">[243]</a> After Cardano, <a href="/wiki/Jakob_Bernoulli" class="mw-redirect" title="Jakob Bernoulli">Jakob Bernoulli</a><a href="#cite_note-249">[e]</a> wrote <a href="/wiki/Ars_Conjectandi" title="Ars Conjectandi">Ars Conjectandi</a>, which is considered a significant event in the history of probability theory. Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability.<a href="#cite_note-Maistrov2014page56-250">[245]</a><a href="#cite_note-Tabak2014page37-251">[246]</a> But despite some renowned mathematicians contributing to probability theory, such as <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>, <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a>, <a href="/wiki/Carl_Gauss" class="mw-redirect" title="Carl Gauss">Carl Gauss</a>, <a href="/wiki/Sim%C3%A9on_Poisson" class="mw-redirect" title="Siméon Poisson">Siméon Poisson</a> and <a href="/wiki/Pafnuty_Chebyshev" title="Pafnuty Chebyshev">Pafnuty Chebyshev</a>,<a href="#cite_note-Chung1998-252">[247]</a><a href="#cite_note-Bingham2000-253">[248]</a> most of the mathematical community<a href="#cite_note-255">[f]</a> did not consider probability theory to be part of mathematics until the 20th century.<a href="#cite_note-Chung1998-252">[247]</a><a href="#cite_note-BenziBenzi2007-254">[249]</a><a href="#cite_note-Doob1996-256">[250]</a><a href="#cite_note-Cramer1976-257">[251]</a> <div class="mw-heading mw-heading3"><h3 id="Statistical_mechanics">Statistical mechanics</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=39" title="Edit section: Statistical mechanics">edit</a>]</div> In the physical sciences, scientists developed in the 19th century the discipline of <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, where physical systems, such as containers filled with gases, are regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a>, most of the work had little or no randomness.<a href="#cite_note-Truesdell1975page22-258">[252]</a><a href="#cite_note-Brush1967page150-259">[253]</a> This changed in 1859 when <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he modelled the gas particles as moving in random directions at random velocities.<a href="#cite_note-Truesdell1975page31-260">[254]</a><a href="#cite_note-Brush1958-261">[255]</a> The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a> and <a href="/wiki/Josiah_Gibbs" class="mw-redirect" title="Josiah Gibbs">Josiah Gibbs</a>, which would later have an influence on <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s mathematical model for <a href="/wiki/Brownian_movement" class="mw-redirect" title="Brownian movement">Brownian movement</a>.<a href="#cite_note-Brush1968page15-262">[256]</a> <div class="mw-heading mw-heading3"><h3 id="Measure_theory_and_probability_theory">Measure theory and probability theory</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=40" title="Edit section: Measure theory and probability theory">edit</a>]</div> At the <a href="/wiki/International_Congress_of_Mathematicians" title="International Congress of Mathematicians">International Congress of Mathematicians</a> in <a href="/wiki/Paris" title="Paris">Paris</a> in 1900, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> presented a list of <a href="/wiki/Hilbert%27s_problems" title="Hilbert's problems">mathematical problems</a>, where his sixth problem asked for a mathematical treatment of physics and probability involving <a href="/wiki/Axiom" title="Axiom">axioms</a>.<a href="#cite_note-Bingham2000-253">[248]</a> Around the start of the 20th century, mathematicians developed measure theory, a branch of mathematics for studying integrals of mathematical functions, where two of the founders were French mathematicians, <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a> and <a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Émile Borel</a>. In 1925, another French mathematician <a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a> published the first probability book that used ideas from measure theory.<a href="#cite_note-Bingham2000-253">[248]</a> In the 1920s, fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as <a href="/wiki/Sergei_Bernstein" title="Sergei Bernstein">Sergei Bernstein</a>, <a href="/wiki/Aleksandr_Khinchin" title="Aleksandr Khinchin">Aleksandr Khinchin</a>,<a href="#cite_note-263">[g]</a> and <a href="/wiki/Andrei_Kolmogorov" class="mw-redirect" title="Andrei Kolmogorov">Andrei Kolmogorov</a>.<a href="#cite_note-Cramer1976-257">[251]</a> Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory.<a href="#cite_note-KendallBatchelor1990page33-264">[257]</a> In the early 1930s, Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as <a href="/wiki/Eugene_Slutsky" class="mw-redirect" title="Eugene Slutsky">Eugene Slutsky</a> and <a href="/wiki/Nikolai_Smirnov_(mathematician)" title="Nikolai Smirnov (mathematician)">Nikolai Smirnov</a>,<a href="#cite_note-Vere-Jones2006page1-265">[258]</a> and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.<a href="#cite_note-Doob1934-64">[63]</a><a href="#cite_note-Vere-Jones2006page4-266">[259]</a><a href="#cite_note-268">[h]</a> <div class="mw-heading mw-heading3"><h3 id="Birth_of_modern_probability_theory">Birth of modern probability theory</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=41" title="Edit section: Birth of modern probability theory">edit</a>]</div> In 1933, Andrei Kolmogorov published in German, his book on the foundations of probability theory titled Grundbegriffe der Wahrscheinlichkeitsrechnung,<a href="#cite_note-269">[i]</a> where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.<a href="#cite_note-Bingham2000-253">[248]</a><a href="#cite_note-Cramer1976-257">[251]</a> After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as <a href="/wiki/Joseph_Doob" class="mw-redirect" title="Joseph Doob">Joseph Doob</a>, <a href="/wiki/William_Feller" title="William Feller">William Feller</a>, <a href="/wiki/Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Maurice Fréchet</a>, <a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a>, <a href="/wiki/Wolfgang_Doeblin" title="Wolfgang Doeblin">Wolfgang Doeblin</a>, and <a href="/wiki/Harald_Cram%C3%A9r" title="Harald Cramér">Harald Cramér</a>.<a href="#cite_note-Bingham2000-253">[248]</a><a href="#cite_note-Cramer1976-257">[251]</a> Decades later, Cramér referred to the 1930s as the "heroic period of mathematical probability theory".<a href="#cite_note-Cramer1976-257">[251]</a> <a href="/wiki/World_War_II" title="World War II">World War II</a> greatly interrupted the development of probability theory, causing, for example, the migration of Feller from <a href="/wiki/Sweden" title="Sweden">Sweden</a> to the <a href="/wiki/United_States" title="United States">United States of America</a><a href="#cite_note-Cramer1976-257">[251]</a> and the death of Doeblin, considered now a pioneer in stochastic processes.<a href="#cite_note-Lindvall1991-270">[261]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Joseph_Doob.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Joseph_Doob.jpg/220px-Joseph_Doob.jpg" decoding="async" width="220" height="326" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/96/Joseph_Doob.jpg 1.5x" data-file-width="270" data-file-height="400" /></a><figcaption>Mathematician <a href="/wiki/Joseph_Doob" class="mw-redirect" title="Joseph Doob">Joseph Doob</a> did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales.<a href="#cite_note-Getoor2009-271">[262]</a><a href="#cite_note-Snell2005-267">[260]</a> His book Stochastic Processes is considered highly influential in the field of probability theory.<a href="#cite_note-Bingham2005-272">[263]</a> </figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Stochastic_processes_after_World_War_II">Stochastic processes after World War II</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=42" title="Edit section: Stochastic processes after World War II">edit</a>]</div> After World War II, the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas.<a href="#cite_note-Cramer1976-257">[251]</a><a href="#cite_note-Meyer2009-273">[264]</a> Starting in the 1940s, <a href="/wiki/Kiyosi_It%C3%B4" title="Kiyosi Itô">Kiyosi Itô</a> published papers developing the field of <a href="/wiki/Stochastic_calculus" title="Stochastic calculus">stochastic calculus</a>, which involves stochastic <a href="/wiki/Integrals" class="mw-redirect" title="Integrals">integrals</a> and stochastic <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> based on the Wiener or Brownian motion process.<a href="#cite_note-Ito1998Prize-274">[265]</a> Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of <a href="/wiki/Potential_theory" title="Potential theory">potential theory</a>, with early ideas by <a href="/wiki/Shizuo_Kakutani" title="Shizuo Kakutani">Shizuo Kakutani</a> and then later work by Joseph Doob.<a href="#cite_note-Meyer2009-273">[264]</a> Further work, considered pioneering, was done by <a href="/wiki/Gilbert_Hunt" title="Gilbert Hunt">Gilbert Hunt</a> in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of Lévy processes and led to more interest in studying Markov processes with methods developed by Itô.<a href="#cite_note-JarrowProtter2004-22">[21]</a><a href="#cite_note-Bertoin1998pageVIIIandIX-275">[266]</a><a href="#cite_note-Steele2012page176-276">[267]</a> In 1953, Doob published his book Stochastic processes, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.<a href="#cite_note-Meyer2009-273">[264]</a> <a href="#cite_note-Bingham2005-272">[263]</a> Doob also chiefly developed the theory of martingales, with later substantial contributions by <a href="/wiki/Paul-Andr%C3%A9_Meyer" title="Paul-André Meyer">Paul-André Meyer</a>. Earlier work had been carried out by <a href="/wiki/Sergei_Bernstein" title="Sergei Bernstein">Sergei Bernstein</a>, <a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a> and <a href="/wiki/Jean_Ville" title="Jean Ville">Jean Ville</a>, the latter adopting the term martingale for the stochastic process.<a href="#cite_note-HallHeyde2014page1-277">[268]</a><a href="#cite_note-Dynkin1989-278">[269]</a> Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.<a href="#cite_note-Meyer2009-273">[264]</a> Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations.<a href="#cite_note-Meyer2009-273">[264]</a> The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and 1970s, fundamental work was done by Alexander Wentzell in the Soviet Union and <a href="/wiki/Monroe_D._Donsker" title="Monroe D. Donsker">Monroe D. Donsker</a> and <a href="/wiki/Srinivasa_Varadhan" class="mw-redirect" title="Srinivasa Varadhan">Srinivasa Varadhan</a> in the United States of America,<a href="#cite_note-Ellis1995page98-279">[270]</a> which would later result in Varadhan winning the 2007 Abel Prize.<a href="#cite_note-RaussenSkau2008-280">[271]</a> In the 1990s and 2000s the theories of <a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a><a href="#cite_note-HenkelKarevski2012page113-281">[272]</a> and <a href="/wiki/Rough_paths" class="mw-redirect" title="Rough paths">rough paths</a><a href="#cite_note-FrizVictoir2010page571-143">[142]</a> were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in <a href="/wiki/Fields_Medal" title="Fields Medal">Fields Medals</a> being awarded to <a href="/wiki/Wendelin_Werner" title="Wendelin Werner">Wendelin Werner</a><a href="#cite_note-Werner2004Fields-282">[273]</a> in 2008 and to <a href="/wiki/Martin_Hairer" title="Martin Hairer">Martin Hairer</a> in 2014.<a href="#cite_note-Hairer2004Fields-283">[274]</a> The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.<a href="#cite_note-BlathImkeller2011-46">[45]</a><a href="#cite_note-Applebaum2004page1336-228">[225]</a> <div class="mw-heading mw-heading3"><h3 id="Discoveries_of_specific_stochastic_processes">Discoveries of specific stochastic processes</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=43" title="Edit section: Discoveries of specific stochastic processes">edit</a>]</div> Although Khinchin gave mathematical definitions of stochastic processes in the 1930s,<a href="#cite_note-Doob1934-64">[63]</a><a href="#cite_note-Vere-Jones2006page4-266">[259]</a> specific stochastic processes had already been discovered in different settings, such as the Brownian motion process and the Poisson process.<a href="#cite_note-JarrowProtter2004-22">[21]</a><a href="#cite_note-GuttorpThorarinsdottir2012-25">[24]</a> Some families of stochastic processes such as point processes or renewal processes have long and complex histories, stretching back centuries.<a href="#cite_note-DaleyVere-Jones2006chap1-284">[275]</a> <div class="mw-heading mw-heading4"><h4 id="Bernoulli_process_2">Bernoulli process</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=44" title="Edit section: Bernoulli process">edit</a>]</div> The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied.<a href="#cite_note-Florescu2014page301-82">[81]</a> The process is a sequence of independent Bernoulli trials,<a href="#cite_note-BertsekasTsitsiklis2002page273-83">[82]</a> which are named after <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a> who used them to study games of chance, including probability problems proposed and studied earlier by Christiaan Huygens.<a href="#cite_note-Hald2005page226-285">[276]</a> Bernoulli's work, including the Bernoulli process, were published in his book Ars Conjectandi in 1713.<a href="#cite_note-Lebowitz1984-286">[277]</a> <div class="mw-heading mw-heading4"><h4 id="Random_walks">Random walks</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=45" title="Edit section: Random walks">edit</a>]</div> In 1905, <a href="/wiki/Karl_Pearson" title="Karl Pearson">Karl Pearson</a> coined the term random walk while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving random walks.<a href="#cite_note-Weiss2006page1-90">[89]</a><a href="#cite_note-Lebowitz1984-286">[277]</a> For example, the problem known as the Gambler's ruin is based on a simple random walk,<a href="#cite_note-KarlinTaylor2012page49-198">[195]</a><a href="#cite_note-Florescu2014page374-287">[278]</a> and is an example of a random walk with absorbing barriers.<a href="#cite_note-Seneta2006page1-245">[241]</a><a href="#cite_note-Ibe2013page5-288">[279]</a> Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods,<a href="#cite_note-Hald2005page63-289">[280]</a> and then more detailed solutions were presented by Jakob Bernoulli and <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a>.<a href="#cite_note-Hald2005page202-290">[281]</a> For random walks in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional integer <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattices</a>, <a href="/wiki/George_P%C3%B3lya" title="George Pólya">George Pólya</a> published, in 1919 and 1921, work where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.<a href="#cite_note-Florescu2014page385-291">[282]</a><a href="#cite_note-Hughes1995page111-292">[283]</a> <div class="mw-heading mw-heading4"><h4 id="Wiener_process_2">Wiener process</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=46" title="Edit section: Wiener process">edit</a>]</div> The <a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a> or Brownian motion process has its origins in different fields including statistics, finance and physics.<a href="#cite_note-JarrowProtter2004-22">[21]</a> In 1880, Danish astronomer <a href="/wiki/Thorvald_Thiele" class="mw-redirect" title="Thorvald Thiele">Thorvald Thiele</a> wrote a paper on the method of least squares, where he used the process to study the errors of a model in time-series analysis.<a href="#cite_note-Thiele1880-293">[284]</a><a href="#cite_note-Hald1981page1and18-294">[285]</a><a href="#cite_note-Lauritzen1981page319-295">[286]</a> The work is now considered as an early discovery of the statistical method known as <a href="/wiki/Kalman_filtering" class="mw-redirect" title="Kalman filtering">Kalman filtering</a>, but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time.<a href="#cite_note-Lauritzen1981page319-295">[286]</a> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Wiener_Zurich1932.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Wiener_Zurich1932.tif/lossy-page1-200px-Wiener_Zurich1932.tif.jpg" decoding="async" width="200" height="418" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Wiener_Zurich1932.tif/lossy-page1-300px-Wiener_Zurich1932.tif.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Wiener_Zurich1932.tif/lossy-page1-400px-Wiener_Zurich1932.tif.jpg 2x" data-file-width="209" data-file-height="437" /></a><figcaption><a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> gave the first mathematical proof of the existence of the Wiener process. This mathematical object had appeared previously in the work of <a href="/wiki/Thorvald_Thiele" class="mw-redirect" title="Thorvald Thiele">Thorvald Thiele</a>, <a href="/wiki/Louis_Bachelier" title="Louis Bachelier">Louis Bachelier</a>, and <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>.<a href="#cite_note-JarrowProtter2004-22">[21]</a></figcaption></figure> The French mathematician <a href="/wiki/Louis_Bachelier" title="Louis Bachelier">Louis Bachelier</a> used a Wiener process in his 1900 thesis<a href="#cite_note-Bachelier1900a-296">[287]</a><a href="#cite_note-Bachelier1900b-297">[288]</a> in order to model price changes on the <a href="/wiki/Paris_Bourse" class="mw-redirect" title="Paris Bourse">Paris Bourse</a>, a <a href="/wiki/Stock_exchange" title="Stock exchange">stock exchange</a>,<a href="#cite_note-CourtaultKabanov2000-298">[289]</a> without knowing the work of Thiele.<a href="#cite_note-JarrowProtter2004-22">[21]</a> It has been speculated that Bachelier drew ideas from the random walk model of <a href="/wiki/Jules_Regnault" title="Jules Regnault">Jules Regnault</a>, but Bachelier did not cite him,<a href="#cite_note-Jovanovic2012-299">[290]</a> and Bachelier's thesis is now considered pioneering in the field of financial mathematics.<a href="#cite_note-CourtaultKabanov2000-298">[289]</a><a href="#cite_note-Jovanovic2012-299">[290]</a> It is commonly thought that Bachelier's work gained little attention and was forgotten for decades until it was rediscovered in the 1950s by the <a href="/wiki/Leonard_Savage" class="mw-redirect" title="Leonard Savage">Leonard Savage</a>, and then become more popular after Bachelier's thesis was translated into English in 1964. But the work was never forgotten in the mathematical community, as Bachelier published a book in 1912 detailing his ideas,<a href="#cite_note-Jovanovic2012-299">[290]</a> which was cited by mathematicians including Doob, Feller<a href="#cite_note-Jovanovic2012-299">[290]</a> and Kolmogorov.<a href="#cite_note-JarrowProtter2004-22">[21]</a> The book continued to be cited, but then starting in the 1960s, the original thesis by Bachelier began to be cited more than his book when economists started citing Bachelier's work.<a href="#cite_note-Jovanovic2012-299">[290]</a> In 1905, <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a> published a paper where he studied the physical observation of Brownian motion or movement to explain the seemingly random movements of particles in liquids by using ideas from the <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory of gases</a>. Einstein derived a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a>, known as a <a href="/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a>, for describing the probability of finding a particle in a certain region of space. Shortly after Einstein's first paper on Brownian movement, <a href="/wiki/Marian_Smoluchowski" title="Marian Smoluchowski">Marian Smoluchowski</a> published work where he cited Einstein, but wrote that he had independently derived the equivalent results by using a different method.<a href="#cite_note-Brush1968page25-300">[291]</a> Einstein's work, as well as experimental results obtained by <a href="/wiki/Jean_Perrin" class="mw-redirect" title="Jean Perrin">Jean Perrin</a>, later inspired Norbert Wiener in the 1920s<a href="#cite_note-Brush1968page30-301">[292]</a> to use a type of measure theory, developed by <a href="/wiki/Percy_Daniell" class="mw-redirect" title="Percy Daniell">Percy Daniell</a>, and Fourier analysis to prove the existence of the Wiener process as a mathematical object.<a href="#cite_note-JarrowProtter2004-22">[21]</a> <div class="mw-heading mw-heading4"><h4 id="Poisson_process_2">Poisson process</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=47" title="Edit section: Poisson process">edit</a>]</div> The Poisson process is named after <a href="/wiki/Sim%C3%A9on_Poisson" class="mw-redirect" title="Siméon Poisson">Siméon Poisson</a>, due to its definition involving the <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a>, but Poisson never studied the process.<a href="#cite_note-Stirzaker2000-23">[22]</a><a href="#cite_note-DaleyVere-Jones2006page8-302">[293]</a> There are a number of claims for early uses or discoveries of the Poisson process.<a href="#cite_note-Stirzaker2000-23">[22]</a><a href="#cite_note-GuttorpThorarinsdottir2012-25">[24]</a> At the beginning of the 20th century, the Poisson process would arise independently in different situations.<a href="#cite_note-Stirzaker2000-23">[22]</a><a href="#cite_note-GuttorpThorarinsdottir2012-25">[24]</a> In Sweden 1903, <a href="/wiki/Filip_Lundberg" title="Filip Lundberg">Filip Lundberg</a> published a <a href="/wiki/Thesis" title="Thesis">thesis</a> containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.<a href="#cite_note-EmbrechtsFrey2001page367-303">[294]</a><a href="#cite_note-Cramér1969-304">[295]</a> Another discovery occurred in <a href="/wiki/Denmark" title="Denmark">Denmark</a> in 1909 when <a href="/wiki/A.K._Erlang" class="mw-redirect" title="A.K. Erlang">A.K. Erlang</a> derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.<a href="#cite_note-Stirzaker2000-23">[22]</a> In 1910, <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a> and <a href="/wiki/Hans_Geiger" title="Hans Geiger">Hans Geiger</a> published experimental results on counting alpha particles. Motivated by their work, <a href="/wiki/Harry_Bateman" title="Harry Bateman">Harry Bateman</a> studied the counting problem and derived Poisson probabilities as a solution to a family of differential equations, resulting in the independent discovery of the Poisson process.<a href="#cite_note-Stirzaker2000-23">[22]</a> After this time there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.<a href="#cite_note-Stirzaker2000-23">[22]</a> <div class="mw-heading mw-heading4"><h4 id="Markov_processes">Markov processes</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=48" title="Edit section: Markov processes">edit</a>]</div> Markov processes and Markov chains are named after <a href="/wiki/Andrey_Markov" title="Andrey Markov">Andrey Markov</a> who studied Markov chains in the early 20th century. Markov was interested in studying an extension of independent random sequences. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a <a href="/wiki/Weak_law_of_large_numbers" class="mw-redirect" title="Weak law of large numbers">weak law of large numbers</a> without the independence assumption,<a href="#cite_note-GrinsteadSnell1997page464-305">[296]</a><a href="#cite_note-Bremaud2013pageIX-306">[297]</a><a href="#cite_note-Hayes2013-307">[298]</a> which had been commonly regarded as a requirement for such mathematical laws to hold.<a href="#cite_note-Hayes2013-307">[298]</a> Markov later used Markov chains to study the distribution of vowels in <a href="/wiki/Eugene_Onegin" title="Eugene Onegin">Eugene Onegin</a>, written by <a href="/wiki/Alexander_Pushkin" title="Alexander Pushkin">Alexander Pushkin</a>, and proved a <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> for such chains. In 1912, Poincaré studied Markov chains on <a href="/wiki/Finite_group" title="Finite group">finite groups</a> with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by <a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Paul</a> and <a href="/wiki/Tatyana_Ehrenfest" title="Tatyana Ehrenfest">Tatyana Ehrenfest</a> in 1907, and a branching process, introduced by <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a> and <a href="/wiki/Henry_William_Watson" title="Henry William Watson">Henry William Watson</a> in 1873, preceding the work of Markov.<a href="#cite_note-GrinsteadSnell1997page464-305">[296]</a><a href="#cite_note-Bremaud2013pageIX-306">[297]</a> After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by <a href="/wiki/Ir%C3%A9n%C3%A9e-Jules_Bienaym%C3%A9" title="Irénée-Jules Bienaymé">Irénée-Jules Bienaymé</a>.<a href="#cite_note-Seneta1998-308">[299]</a> Starting in 1928, <a href="/wiki/Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Maurice Fréchet</a> became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.<a href="#cite_note-GrinsteadSnell1997page464-305">[296]</a><a href="#cite_note-BruHertz2001-309">[300]</a> <a href="/wiki/Andrei_Kolmogorov" class="mw-redirect" title="Andrei Kolmogorov">Andrei Kolmogorov</a> developed in a 1931 paper a large part of the early theory of continuous-time Markov processes.<a href="#cite_note-Cramer1976-257">[251]</a><a href="#cite_note-KendallBatchelor1990page33-264">[257]</a> Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a>'s work on Einstein's model of Brownian movement.<a href="#cite_note-KendallBatchelor1990page33-264">[257]</a><a href="#cite_note-BarbutLocker2016page5-310">[301]</a> He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.<a href="#cite_note-KendallBatchelor1990page33-264">[257]</a><a href="#cite_note-Skorokhod2005page146-311">[302]</a> Independent of Kolmogorov's work, <a href="/wiki/Sydney_Chapman_(mathematician)" title="Sydney Chapman (mathematician)">Sydney Chapman</a> derived in a 1928 paper an equation, now called the <a href="/wiki/Chapman%E2%80%93Kolmogorov_equation" title="Chapman–Kolmogorov equation">Chapman–Kolmogorov equation</a>, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.<a href="#cite_note-Bernstein2005-312">[303]</a> The differential equations are now called the Kolmogorov equations<a href="#cite_note-Anderson2012pageVII-313">[304]</a> or the Kolmogorov–Chapman equations.<a href="#cite_note-KendallBatchelor1990page57-314">[305]</a> Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s.<a href="#cite_note-Cramer1976-257">[251]</a> <div class="mw-heading mw-heading4"><h4 id="Lévy_processes">Lévy processes</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=49" title="Edit section: Lévy processes">edit</a>]</div> Lévy processes such as the Wiener process and the Poisson process (on the real line) are named after Paul Lévy who started studying them in the 1930s,<a href="#cite_note-Applebaum2004page1336-228">[225]</a> but they have connections to <a href="/wiki/Infinitely_divisible_distribution" class="mw-redirect" title="Infinitely divisible distribution">infinitely divisible distributions</a> going back to the 1920s.<a href="#cite_note-Bertoin1998pageVIII-227">[224]</a> In a 1932 paper, Kolmogorov derived a <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> for random variables associated with Lévy processes. This result was later derived under more general conditions by Lévy in 1934, and then Khinchin independently gave an alternative form for this characteristic function in 1937.<a href="#cite_note-Cramer1976-257">[251]</a><a href="#cite_note-ApplebaumBook2004page67-315">[306]</a> In addition to Lévy, Khinchin and Kolomogrov, early fundamental contributions to the theory of Lévy processes were made by <a href="/wiki/Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a> and <a href="/wiki/Kiyosi_It%C3%B4" title="Kiyosi Itô">Kiyosi Itô</a>.<a href="#cite_note-Bertoin1998pageVIII-227">[224]</a> <div class="mw-heading mw-heading2"><h2 id="Mathematical_construction">Mathematical construction</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=50" title="Edit section: Mathematical construction">edit</a>]</div> In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically.<a href="#cite_note-Rosenthal2006page177-58">[57]</a> There are two main approaches for constructing a stochastic process. One approach involves considering a measurable space of functions, defining a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.<a href="#cite_note-Adler2010page13-316">[307]</a> Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using <a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov's existence theorem</a><a href="#cite_note-320">[j]</a> to prove a corresponding stochastic process exists.<a href="#cite_note-Rosenthal2006page177-58">[57]</a><a href="#cite_note-Adler2010page13-316">[307]</a> This theorem, which is an existence theorem for measures on infinite product spaces,<a href="#cite_note-Durrett2010page410-321">[311]</a> says that if any finite-dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite-dimensional distributions.<a href="#cite_note-Rosenthal2006page177-58">[57]</a> <div class="mw-heading mw-heading3"><h3 id="Construction_issues">Construction issues</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=51" title="Edit section: Construction issues">edit</a>]</div> When constructing continuous-time stochastic processes certain mathematical difficulties arise, due to the uncountable index sets, which do not occur with discrete-time processes.<a href="#cite_note-KloedenPlaten2013page63-59">[58]</a><a href="#cite_note-Khoshnevisan2006page153-60">[59]</a> One problem is that it is possible to have more than one stochastic process with the same finite-dimensional distributions. For example, both the left-continuous modification and the right-continuous modification of a Poisson process have the same finite-dimensional distributions.<a href="#cite_note-Billingsley2008page493to494-322">[312]</a> This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.<a href="#cite_note-Adler2010page13-316">[307]</a><a href="#cite_note-Borovkov2013page529-323">[313]</a> Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined.<a href="#cite_note-Ito2006page32-170">[168]</a> For example, the supremum of a stochastic process or random field is not necessarily a well-defined random variable.<a href="#cite_note-AdlerTaylor2009page7-31">[30]</a><a href="#cite_note-Khoshnevisan2006page153-60">[59]</a> For a continuous-time stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, other characteristics that depend on an uncountable number of points of the index set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> include:<a href="#cite_note-Ito2006page32-170">[168]</a> <ul><li>a sample function of a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}">;</li> <li>a sample function of a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a <a href="/wiki/Bounded_function" title="Bounded function">bounded function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}">; and</li> <li>a sample function of a stochastic process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is an <a href="/wiki/Increasing_function" class="mw-redirect" title="Increasing function">increasing function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}">.</li></ul> where the symbol ∈ can be read "a member of the set", as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> a member of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">. To overcome the two difficulties described above, i.e., "more than one..." and "functionals of...", different assumptions and approaches are possible.<a href="#cite_note-Asmussen2003page408-70">[69]</a> <div class="mw-heading mw-heading3"><h3 id="Resolving_construction_issues">Resolving construction issues</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=52" title="Edit section: Resolving construction issues">edit</a>]</div> One approach for avoiding mathematical construction issues of stochastic processes, proposed by <a href="/wiki/Joseph_Doob" class="mw-redirect" title="Joseph Doob">Joseph Doob</a>, is to assume that the stochastic process is separable.<a href="#cite_note-AthreyaLahiri2006page221-324">[314]</a> Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set.<a href="#cite_note-AdlerTaylor2009page14-325">[315]</a> Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied.<a href="#cite_note-Ito2006page32-170">[168]</a><a href="#cite_note-AdlerTaylor2009page14-325">[315]</a> Another approach is possible, originally developed by <a href="/wiki/Anatoliy_Skorokhod" title="Anatoliy Skorokhod">Anatoliy Skorokhod</a> and <a href="/wiki/Andrei_Kolmogorov" class="mw-redirect" title="Andrei Kolmogorov">Andrei Kolmogorov</a>,<a href="#cite_note-AthreyaLahiri2006page211-326">[316]</a> for a continuous-time stochastic process with any metric space as its state space. For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits. This approach is now more used than the separability assumption,<a href="#cite_note-Asmussen2003page408-70">[69]</a><a href="#cite_note-Getoor2009-271">[262]</a> but such a stochastic process based on this approach will be automatically separable.<a href="#cite_note-Borovkov2013page536-327">[317]</a> Although less used, the separability assumption is considered more general because every stochastic process has a separable version.<a href="#cite_note-Getoor2009-271">[262]</a> It is also used when it is not possible to construct a stochastic process in a Skorokhod space.<a href="#cite_note-Borovkov2013page535-175">[173]</a> For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional Euclidean space.<a href="#cite_note-AdlerTaylor2009page7-31">[30]</a><a href="#cite_note-Yakir2013page5-328">[318]</a> <div class="mw-heading mw-heading2"><h2 id="Application">Application</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=53" title="Edit section: Application">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Applications_in_Finance">Applications in Finance</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=54" title="Edit section: Applications in Finance">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Black-Scholes_Model">Black-Scholes Model</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=55" title="Edit section: Black-Scholes Model">edit</a>]</div> One of the most famous applications of stochastic processes in finance is the <a href="/wiki/Black-Scholes_model" class="mw-redirect" title="Black-Scholes model">Black-Scholes model</a> for option pricing. Developed by <a href="/wiki/Fischer_Black" title="Fischer Black">Fischer Black</a>, <a href="/wiki/Myron_Scholes" title="Myron Scholes">Myron Scholes</a>, and <a href="/wiki/Robert_Solow" title="Robert Solow">Robert Solow</a>, this model uses <a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric Brownian motion</a>, a specific type of stochastic process, to describe the dynamics of asset prices.<a href="#cite_note-329">[319]</a><a href="#cite_note-330">[320]</a> The model assumes that the price of a stock follows a continuous-time stochastic process and provides a closed-form solution for pricing European-style options. The Black-Scholes formula has had a profound impact on financial markets, forming the basis for much of modern options trading. The key assumption of the Black-Scholes model is that the price of a financial asset, such as a stock, follows a <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a>, with its continuous returns following a normal distribution. Although the model has limitations, such as the assumption of constant volatility, it remains widely used due to its simplicity and practical relevance. <div class="mw-heading mw-heading4"><h4 id="Stochastic_Volatility_Models">Stochastic Volatility Models</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=56" title="Edit section: Stochastic Volatility Models">edit</a>]</div> Another significant application of stochastic processes in finance is in <a href="/wiki/Stochastic_volatility" title="Stochastic volatility">stochastic volatility models</a>, which aim to capture the time-varying nature of market volatility. The <a href="/wiki/Heston_model" title="Heston model">Heston model</a><a href="#cite_note-331">[321]</a> is a popular example, allowing for the volatility of asset prices to follow its own stochastic process. Unlike the Black-Scholes model, which assumes constant volatility, stochastic volatility models provide a more flexible framework for modeling market dynamics, particularly during periods of high uncertainty or market stress. <div class="mw-heading mw-heading3"><h3 id="Applications_in_Biology">Applications in Biology</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=57" title="Edit section: Applications in Biology">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Population_Dynamics">Population Dynamics</h4>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=58" title="Edit section: Population Dynamics">edit</a>]</div> One of the primary applications of stochastic processes in biology is in <a href="/wiki/Population_dynamics" title="Population dynamics">population dynamics</a>. In contrast to <a href="/wiki/Deterministic_model" class="mw-redirect" title="Deterministic model">deterministic models</a>, which assume that populations change in predictable ways, stochastic models account for the inherent randomness in births, deaths, and migration. The <a href="/wiki/Birth-death_process" class="mw-redirect" title="Birth-death process">birth-death process</a>,<a href="#cite_note-332">[322]</a> a simple stochastic model, describes how populations fluctuate over time due to random births and deaths. These models are particularly important when dealing with small populations, where random events can have large impacts, such as in the case of endangered species or small microbial populations. Another example is the <a href="/wiki/Branching_process" title="Branching process">branching process</a>,<a href="#cite_note-333">[323]</a> which models the growth of a population where each individual reproduces independently. The branching process is often used to describe population extinction or explosion, particularly in epidemiology, where it can model the spread of infectious diseases within a population. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=59" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of stochastic processes topics</a></li> <li><a href="/wiki/Covariance_function" title="Covariance function">Covariance function</a></li> <li><a href="/wiki/Deterministic_system" title="Deterministic system">Deterministic system</a></li> <li><a href="/wiki/Dynamics_of_Markovian_particles" title="Dynamics of Markovian particles">Dynamics of Markovian particles</a></li> <li><a href="/wiki/Entropy_rate" title="Entropy rate">Entropy rate</a> (for a stochastic process)</li> <li><a href="/wiki/Ergodic_process" title="Ergodic process">Ergodic process</a></li> <li><a href="/wiki/Gillespie_algorithm" title="Gillespie algorithm">Gillespie algorithm</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle system</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Stochastic_cellular_automaton" title="Stochastic cellular automaton">Stochastic cellular automaton</a></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary process</a></li> <li><a href="/wiki/Statistical_model" title="Statistical model">Statistical model</a></li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic calculus</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Stochastic control</a></li> <li><a href="/wiki/Stochastic_parrot" title="Stochastic parrot">Stochastic parrot</a></li> <li><a href="/wiki/Stochastic_processes_and_boundary_value_problems" title="Stochastic processes and boundary value problems">Stochastic processes and boundary value problems</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=60" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-21"><a href="#cite_ref-21">^</a> The term Brownian motion can refer to the physical process, also known as Brownian movement, and the stochastic process, a mathematical object, but to avoid ambiguity this article uses the terms Brownian motion process or Wiener process for the latter in a style similar to, for example, <a href="/wiki/Iosif_Gikhman" title="Iosif Gikhman">Gikhman</a> and <a href="/wiki/Anatoliy_Skorokhod" title="Anatoliy Skorokhod">Skorokhod</a><a href="#cite_note-GikhmanSkorokhod1969-19">[19]</a> or Rosenblatt.<a href="#cite_note-Rosenblatt1962-20">[20]</a> </li> <li id="cite_note-169"><a href="#cite_ref-169">^</a> The term "separable" appears twice here with two different meanings, where the first meaning is from probability and the second from topology and analysis. For a stochastic process to be separable (in a probabilistic sense), its index set must be a separable space (in a topological or analytic sense), in addition to other conditions.<a href="#cite_note-Skorokhod2005page93-137">[136]</a> </li> <li id="cite_note-176"><a href="#cite_ref-176">^</a> The definition of separability for a continuous-time real-valued stochastic process can be stated in other ways.<a href="#cite_note-Billingsley2008page526-174">[172]</a><a href="#cite_note-Borovkov2013page535-175">[173]</a> </li> <li id="cite_note-239"><a href="#cite_ref-239">^</a> In the context of point processes, the term "state space" can mean the space on which the point process is defined such as the real line,<a href="#cite_note-Kingman1992page8-237">[234]</a><a href="#cite_note-MollerWaagepetersen2003page7-238">[235]</a> which corresponds to the index set in stochastic process terminology. </li> <li id="cite_note-249"><a href="#cite_ref-249">^</a> Also known as James or Jacques Bernoulli.<a href="#cite_note-Hald2005page221-248">[244]</a> </li> <li id="cite_note-255"><a href="#cite_ref-255">^</a> It has been remarked that a notable exception was the St Petersburg School in Russia, where mathematicians led by Chebyshev studied probability theory.<a href="#cite_note-BenziBenzi2007-254">[249]</a> </li> <li id="cite_note-263"><a href="#cite_ref-263">^</a> The name Khinchin is also written in (or transliterated into) English as Khintchine.<a href="#cite_note-Doob1934-64">[63]</a> </li> <li id="cite_note-268"><a href="#cite_ref-268">^</a> Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.<a href="#cite_note-Snell2005-267">[260]</a> </li> <li id="cite_note-269"><a href="#cite_ref-269">^</a> Later translated into English and published in 1950 as Foundations of the Theory of Probability<a href="#cite_note-Bingham2000-253">[248]</a> </li> <li id="cite_note-320"><a href="#cite_ref-320">^</a> The theorem has other names including Kolmogorov's consistency theorem,<a href="#cite_note-AthreyaLahiri2006-317">[308]</a> Kolmogorov's extension theorem<a href="#cite_note-Øksendal2003page11-318">[309]</a> or the Daniell–Kolmogorov theorem.<a href="#cite_note-Williams1991page124-319">[310]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=61" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-doob1953stochasticP46to47-1">^ <a href="#cite_ref-doob1953stochasticP46to47_1-0">a</a> <a href="#cite_ref-doob1953stochasticP46to47_1-1">b</a> <a href="#cite_ref-doob1953stochasticP46to47_1-2">c</a> <a href="#cite_ref-doob1953stochasticP46to47_1-3">d</a> <a href="#cite_ref-doob1953stochasticP46to47_1-4">e</a> <a href="#cite_ref-doob1953stochasticP46to47_1-5">f</a> <a href="#cite_ref-doob1953stochasticP46to47_1-6">g</a> <a href="#cite_ref-doob1953stochasticP46to47_1-7">h</a> <a href="#cite_ref-doob1953stochasticP46to47_1-8">i</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJoseph_L._Doob1990" class="citation book cs1">Joseph L. Doob (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7Bu8jgEACAAJ">Stochastic processes</a>. Wiley. pp. 46, 47.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+processes&rft.pages=46%2C+47&rft.pub=Wiley&rft.date=1990&rft.au=Joseph+L.+Doob&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7Bu8jgEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-RogersWilliams2000page1-2">^ <a href="#cite_ref-RogersWilliams2000page1_2-0">a</a> <a href="#cite_ref-RogersWilliams2000page1_2-1">b</a> <a href="#cite_ref-RogersWilliams2000page1_2-2">c</a> <a href="#cite_ref-RogersWilliams2000page1_2-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFL._C._G._RogersDavid_Williams2000" class="citation book cs1">L. C. G. 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Cambridge University Press. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-71749-7" title="Special:BookSources/978-1-107-71749-7"><bdi>978-1-107-71749-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Diffusions%2C+Markov+Processes%2C+and+Martingales%3A+Volume+1%2C+Foundations&rft.pages=1&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-1-107-71749-7&rft.au=L.+C.+G.+Rogers&rft.au=David+Williams&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DW0ydAgAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Steele2012page29-3">^ <a href="#cite_ref-Steele2012page29_3-0">a</a> <a href="#cite_ref-Steele2012page29_3-1">b</a> <a href="#cite_ref-Steele2012page29_3-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ._Michael_Steele2012" class="citation book cs1">J. Michael Steele (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4">Stochastic Calculus and Financial Applications</a>. Springer Science & Business Media. p. 29. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4684-9305-4" title="Special:BookSources/978-1-4684-9305-4"><bdi>978-1-4684-9305-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Calculus+and+Financial+Applications&rft.pages=29&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4684-9305-4&rft.au=J.+Michael+Steele&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfsgkBAAAQBAJ%26pg%3DPR4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Parzen1999-4">^ <a href="#cite_ref-Parzen1999_4-0">a</a> <a href="#cite_ref-Parzen1999_4-1">b</a> <a href="#cite_ref-Parzen1999_4-2">c</a> <a href="#cite_ref-Parzen1999_4-3">d</a> <a href="#cite_ref-Parzen1999_4-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmanuel_Parzen2015" class="citation book cs1">Emanuel Parzen (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0mB2CQAAQBAJ">Stochastic Processes</a>. Courier Dover Publications. pp. 7, 8. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-79688-8" title="Special:BookSources/978-0-486-79688-8"><bdi>978-0-486-79688-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Processes&rft.pages=7%2C+8&rft.pub=Courier+Dover+Publications&rft.date=2015&rft.isbn=978-0-486-79688-8&rft.au=Emanuel+Parzen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0mB2CQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-GikhmanSkorokhod1969page1-5">^ <a href="#cite_ref-GikhmanSkorokhod1969page1_5-0">a</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-1">b</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-2">c</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-3">d</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-4">e</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-5">f</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-6">g</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-7">h</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-8">i</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-9">j</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-10">k</a> <a href="#cite_ref-GikhmanSkorokhod1969page1_5-11">l</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIosif_Ilyich_GikhmanAnatoly_Vladimirovich_Skorokhod1969" class="citation book cs1">Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q0lo91imeD0C">Introduction to the Theory of Random Processes</a>. Courier Corporation. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-69387-3" title="Special:BookSources/978-0-486-69387-3"><bdi>978-0-486-69387-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Random+Processes&rft.pages=1&rft.pub=Courier+Corporation&rft.date=1969&rft.isbn=978-0-486-69387-3&rft.au=Iosif+Ilyich+Gikhman&rft.au=Anatoly+Vladimirovich+Skorokhod&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dq0lo91imeD0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Bressloff2014-6"><a href="#cite_ref-Bressloff2014_6-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBressloff2014" class="citation book cs1"><a href="/wiki/Paul_Bressloff" title="Paul Bressloff">Bressloff, Paul C.</a> (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SwZYBAAAQBAJ">Stochastic Processes in Cell Biology</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-08488-6" title="Special:BookSources/978-3-319-08488-6"><bdi>978-3-319-08488-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Processes+in+Cell+Biology&rft.pub=Springer&rft.date=2014&rft.isbn=978-3-319-08488-6&rft.aulast=Bressloff&rft.aufirst=Paul+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSwZYBAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Kampen2011-7"><a href="#cite_ref-Kampen2011_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Kampen2011" class="citation book cs1">Van Kampen, N. G. 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Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-71749-7" title="Special:BookSources/978-1-107-71749-7"><bdi>978-1-107-71749-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Diffusions%2C+Markov+Processes%2C+and+Martingales%3A+Volume+1%2C+Foundations&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-1-107-71749-7&rft.au=L.+C.+G.+Rogers&rft.au=David+Williams&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DW0ydAgAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-ApplebaumBook2004-35"><a href="#cite_ref-ApplebaumBook2004_35-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Applebaum2004" class="citation book cs1">David Applebaum (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q7eDUjdJxIkC">Lévy Processes and Stochastic Calculus</a>. 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Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-24939-6" title="Special:BookSources/978-3-642-24939-6"><bdi>978-3-642-24939-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Gaussian+Processes&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-3-642-24939-6&rft.au=Mikhail+Lifshits&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D03m2UxI-UYMC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Adler2010-37"><a href="#cite_ref-Adler2010_37-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_J._Adler2010" class="citation book cs1">Robert J. 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SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-693-1" title="Special:BookSources/978-0-89871-693-1"><bdi>978-0-89871-693-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Random+Fields&rft.pub=SIAM&rft.date=2010&rft.isbn=978-0-89871-693-1&rft.au=Robert+J.+Adler&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DryejJmJAj28C%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-KarlinTaylor2012-38"><a href="#cite_ref-KarlinTaylor2012_38-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamuel_KarlinHoward_E._Taylor2012" class="citation book cs1">Samuel Karlin; Howard E. 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Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-21337-8" title="Special:BookSources/978-0-387-21337-8"><bdi>978-0-387-21337-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Point+Processes%3A+Volume+II%3A+General+Theory+and+Structure&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-0-387-21337-8&rft.au=D.J.+Daley&rft.au=David+Vere-Jones&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnPENXKw5kwcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Billingsley2008-42"><a href="#cite_ref-Billingsley2008_42-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPatrick_Billingsley2008" class="citation book cs1">Patrick Billingsley (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QyXqOXyxEeIC">Probability and Measure</a>. 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European Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-072-2" title="Special:BookSources/978-3-03719-072-2"><bdi>978-3-03719-072-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Surveys+in+Stochastic+Processes&rft.pub=European+Mathematical+Society&rft.date=2011&rft.isbn=978-3-03719-072-2&rft.au=Jochen+Blath&rft.au=Peter+Imkeller&rft.au=Sylvie+Roelly&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCyK6KAjwdYkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Talagrand2014-47"><a href="#cite_ref-Talagrand2014_47-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichel_Talagrand2014" class="citation book cs1">Michel Talagrand (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tfa5BAAAQBAJ&pg=PR4">Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems</a>. 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Cambridge University Press. p. 571. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-48721-4" title="Special:BookSources/978-1-139-48721-4"><bdi>978-1-139-48721-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multidimensional+Stochastic+Processes+as+Rough+Paths%3A+Theory+and+Applications&rft.pages=571&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-1-139-48721-4&rft.au=Peter+K.+Friz&rft.au=Nicolas+B.+Victoir&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCVgwLatxfGsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Resnick2013page40-144"><a href="#cite_ref-Resnick2013page40_144-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSidney_I._Resnick2013" class="citation book cs1">Sidney I. 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Springer Science & Business Media. pp. 40–41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-0387-2" title="Special:BookSources/978-1-4612-0387-2"><bdi>978-1-4612-0387-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Adventures+in+Stochastic+Processes&rft.pages=40-41&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4612-0387-2&rft.au=Sidney+I.+Resnick&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVQrpBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Whitt2006page23-145"><a href="#cite_ref-Whitt2006page23_145-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWard_Whitt2006" class="citation book cs1">Ward Whitt (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LkQOBwAAQBAJ&pg=PR5">Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues</a>. Springer Science & Business Media. p. 23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-21748-2" title="Special:BookSources/978-0-387-21748-2"><bdi>978-0-387-21748-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic-Process+Limits%3A+An+Introduction+to+Stochastic-Process+Limits+and+Their+Application+to+Queues&rft.pages=23&rft.pub=Springer+Science+%26+Business+Media&rft.date=2006&rft.isbn=978-0-387-21748-2&rft.au=Ward+Whitt&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLkQOBwAAQBAJ%26pg%3DPR5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-ApplebaumBook2004page4-146"><a href="#cite_ref-ApplebaumBook2004page4_146-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Applebaum2004" class="citation book cs1">David Applebaum (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q7eDUjdJxIkC">Lévy Processes and Stochastic Calculus</a>. 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Springer Science & Business Media. p. 22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4684-9305-4" title="Special:BookSources/978-1-4684-9305-4"><bdi>978-1-4684-9305-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Calculus+and+Financial+Applications&rft.pages=22&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012&rft.isbn=978-1-4684-9305-4&rft.au=J.+Michael+Steele&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfsgkBAAAQBAJ%26pg%3DPR4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-GrimmettStirzaker2001page336-223"><a href="#cite_ref-GrimmettStirzaker2001page336_223-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeoffrey_GrimmettDavid_Stirzaker2001" class="citation book cs1">Geoffrey Grimmett; David Stirzaker (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G3ig-0M4wSIC">Probability and Random Processes</a>. OUP Oxford. p. 336. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-857222-0" title="Special:BookSources/978-0-19-857222-0"><bdi>978-0-19-857222-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Random+Processes&rft.pages=336&rft.pub=OUP+Oxford&rft.date=2001&rft.isbn=978-0-19-857222-0&rft.au=Geoffrey+Grimmett&rft.au=David+Stirzaker&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DG3ig-0M4wSIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-GlassermanKou2006-224"><a href="#cite_ref-GlassermanKou2006_224-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlassermanKou2006" class="citation journal cs1">Glasserman, Paul; Kou, Steven (2006). "A Conversation with Chris Heyde". Statistical Science. 21 (2): 292, 293. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0609294">math/0609294</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006math......9294G">2006math......9294G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2F088342306000000088">10.1214/088342306000000088</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0883-4237">0883-4237</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:62552177">62552177</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistical+Science&rft.atitle=A+Conversation+with+Chris+Heyde&rft.volume=21&rft.issue=2&rft.pages=292%2C+293&rft.date=2006&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A62552177%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2006math......9294G&rft_id=info%3Aarxiv%2Fmath%2F0609294&rft.issn=0883-4237&rft_id=info%3Adoi%2F10.1214%2F088342306000000088&rft.aulast=Glasserman&rft.aufirst=Paul&rft.au=Kou%2C+Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-BaccelliBremaud2013-225"><a href="#cite_ref-BaccelliBremaud2013_225-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrancois_BaccelliPierre_Bremaud2013" class="citation book cs1">Francois Baccelli; Pierre Bremaud (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2">Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences</a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-11657-9" title="Special:BookSources/978-3-662-11657-9"><bdi>978-3-662-11657-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Queueing+Theory%3A+Palm+Martingale+Calculus+and+Stochastic+Recurrences&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-3-662-11657-9&rft.au=Francois+Baccelli&rft.au=Pierre+Bremaud&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDH3pCAAAQBAJ%26pg%3DPR2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-HallHeyde2014pageX-226"><a href="#cite_ref-HallHeyde2014pageX_226-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._HallC._C._Heyde2014" class="citation book cs1">P. Hall; C. C. Heyde (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10">Martingale Limit Theory and Its Application</a>. Elsevier Science. p. x. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4832-6322-9" title="Special:BookSources/978-1-4832-6322-9"><bdi>978-1-4832-6322-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Martingale+Limit+Theory+and+Its+Application&rft.pages=x&rft.pub=Elsevier+Science&rft.date=2014&rft.isbn=978-1-4832-6322-9&rft.au=P.+Hall&rft.au=C.+C.+Heyde&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgqriBQAAQBAJ%26pg%3DPR10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Bertoin1998pageVIII-227">^ <a href="#cite_ref-Bertoin1998pageVIII_227-0">a</a> <a href="#cite_ref-Bertoin1998pageVIII_227-1">b</a> <a href="#cite_ref-Bertoin1998pageVIII_227-2">c</a> <a href="#cite_ref-Bertoin1998pageVIII_227-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJean_Bertoin1998" class="citation book cs1">Jean Bertoin (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8">Lévy Processes</a>. 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Cambridge University Press. p. 19. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83263-2" title="Special:BookSources/978-0-521-83263-2"><bdi>978-0-521-83263-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=L%C3%A9vy+Processes+and+Stochastic+Calculus&rft.pages=19&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-0-521-83263-2&rft.au=David+Applebaum&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dq7eDUjdJxIkC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-ChiuStoyan2013page109-232"><a href="#cite_ref-ChiuStoyan2013page109_232-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSung_Nok_ChiuDietrich_StoyanWilfrid_S._KendallJoseph_Mecke2013" class="citation book cs1">Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. 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Springer Science & Business Media. pp. 1–4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-21564-8" title="Special:BookSources/978-0-387-21564-8"><bdi>978-0-387-21564-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Point+Processes%3A+Volume+I%3A+Elementary+Theory+and+Methods&rft.pages=1-4&rft.pub=Springer+Science+%26+Business+Media&rft.date=2006&rft.isbn=978-0-387-21564-8&rft.au=D.J.+Daley&rft.au=D.+Vere-Jones&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6Sv4BwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Hald2005page226-285"><a href="#cite_ref-Hald2005page226_285-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnders_Hald2005" class="citation book cs1">Anders Hald (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pOQy6-qnVx8C">A History of Probability and Statistics and Their Applications before 1750</a>. 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John Wiley & Sons. p. 63. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-72517-6" title="Special:BookSources/978-0-471-72517-6"><bdi>978-0-471-72517-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Probability+and+Statistics+and+Their+Applications+before+1750&rft.pages=63&rft.pub=John+Wiley+%26+Sons&rft.date=2005&rft.isbn=978-0-471-72517-6&rft.au=Anders+Hald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpOQy6-qnVx8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Hald2005page202-290"><a href="#cite_ref-Hald2005page202_290-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnders_Hald2005" class="citation book cs1">Anders Hald (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pOQy6-qnVx8C">A History of Probability and Statistics and Their Applications before 1750</a>. John Wiley & Sons. p. 202. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-72517-6" title="Special:BookSources/978-0-471-72517-6"><bdi>978-0-471-72517-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Probability+and+Statistics+and+Their+Applications+before+1750&rft.pages=202&rft.pub=John+Wiley+%26+Sons&rft.date=2005&rft.isbn=978-0-471-72517-6&rft.au=Anders+Hald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpOQy6-qnVx8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Florescu2014page385-291"><a href="#cite_ref-Florescu2014page385_291-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIonut_Florescu2014" class="citation book cs1">Ionut Florescu (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22">Probability and Stochastic Processes</a>. 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The European Journal of the History of Economic Thought. 19 (3): 431–451. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F09672567.2010.540343">10.1080/09672567.2010.540343</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0967-2567">0967-2567</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:154003579">154003579</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180721111017/http://r-libre.teluq.ca/1168/1/dissemination%20of%20Louis%20Bachelier_EJHET_R2.pdf">Archived</a> (PDF) from the original on 2018-07-21.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+European+Journal+of+the+History+of+Economic+Thought&rft.atitle=Bachelier%3A+Not+the+forgotten+forerunner+he+has+been+depicted+as.+An+analysis+of+the+dissemination+of+Louis+Bachelier%27s+work+in+economics&rft.volume=19&rft.issue=3&rft.pages=431-451&rft.date=2012&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A154003579%23id-name%3DS2CID&rft.issn=0967-2567&rft_id=info%3Adoi%2F10.1080%2F09672567.2010.540343&rft.aulast=Jovanovic&rft.aufirst=Franck&rft_id=http%3A%2F%2Fr-libre.teluq.ca%2F1168%2F1%2Fdissemination%2520of%2520Louis%2520Bachelier_EJHET_R2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Brush1968page25-300"><a href="#cite_ref-Brush1968page25_300-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrush1968" class="citation journal cs1">Brush, Stephen G. 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Springer London. p. 5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4471-7262-8" title="Special:BookSources/978-1-4471-7262-8"><bdi>978-1-4471-7262-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Paul+L%C3%A9vy+and+Maurice+Fr%C3%A9chet%3A+50+Years+of+Correspondence+in+107+Letters&rft.pages=5&rft.pub=Springer+London&rft.date=2016&rft.isbn=978-1-4471-7262-8&rft.au=Marc+Barbut&rft.au=Bernard+Locker&rft.au=Laurent+Mazliak&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlSz_vQAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Skorokhod2005page146-311"><a href="#cite_ref-Skorokhod2005page146_311-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFValeriy_Skorokhod2005" class="citation book cs1">Valeriy Skorokhod (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dQkYMjRK3fYC">Basic Principles and Applications of Probability Theory</a>. 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American Journal of Physics. 73 (5): 398–396. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005AmJPh..73..395B">2005AmJPh..73..395B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1848117">10.1119/1.1848117</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9505">0002-9505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Bachelier&rft.volume=73&rft.issue=5&rft.pages=398-396&rft.date=2005&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.1848117&rft_id=info%3Abibcode%2F2005AmJPh..73..395B&rft.aulast=Bernstein&rft.aufirst=Jeremy&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Anderson2012pageVII-313"><a href="#cite_ref-Anderson2012pageVII_313-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_J._Anderson2012" class="citation book cs1">William J. 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Bulletin of the London Mathematical Society. 22 (1): 57. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fblms%2F22.1.31">10.1112/blms/22.1.31</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0024-6093">0024-6093</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+London+Mathematical+Society&rft.atitle=Andrei+Nikolaevich+Kolmogorov+%281903%E2%80%931987%29&rft.volume=22&rft.issue=1&rft.pages=57&rft.date=1990&rft_id=info%3Adoi%2F10.1112%2Fblms%2F22.1.31&rft.issn=0024-6093&rft.aulast=Kendall&rft.aufirst=D.+G.&rft.au=Batchelor%2C+G.+K.&rft.au=Bingham%2C+N.+H.&rft.au=Hayman%2C+W.+K.&rft.au=Hyland%2C+J.+M.+E.&rft.au=Lorentz%2C+G.+G.&rft.au=Moffatt%2C+H.+K.&rft.au=Parry%2C+W.&rft.au=Razborov%2C+A.+A.&rft.au=Robinson%2C+C.+A.&rft.au=Whittle%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-ApplebaumBook2004page67-315"><a href="#cite_ref-ApplebaumBook2004page67_315-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Applebaum2004" class="citation book cs1">David Applebaum (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q7eDUjdJxIkC">Lévy Processes and Stochastic Calculus</a>. 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Springer Science & Business Media. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-48116-6" title="Special:BookSources/978-0-387-48116-6"><bdi>978-0-387-48116-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Fields+and+Geometry&rft.pages=14&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=978-0-387-48116-6&rft.au=Robert+J.+Adler&rft.au=Jonathan+E.+Taylor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DR5BGvQ3ejloC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-AthreyaLahiri2006page211-326"><a href="#cite_ref-AthreyaLahiri2006page211_326-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrishna_B._AthreyaSoumendra_N._Lahiri2006" class="citation book cs1">Krishna B. Athreya; Soumendra N. Lahiri (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9tv0taI8l6YC">Measure Theory and Probability Theory</a>. Springer Science & Business Media. p. 211. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-32903-1" title="Special:BookSources/978-0-387-32903-1"><bdi>978-0-387-32903-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Measure+Theory+and+Probability+Theory&rft.pages=211&rft.pub=Springer+Science+%26+Business+Media&rft.date=2006&rft.isbn=978-0-387-32903-1&rft.au=Krishna+B.+Athreya&rft.au=Soumendra+N.+Lahiri&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9tv0taI8l6YC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Borovkov2013page536-327"><a href="#cite_ref-Borovkov2013page536_327-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander_A._Borovkov2013" class="citation book cs1">Alexander A. Borovkov (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hRk_AAAAQBAJ">Probability Theory</a>. Springer Science & Business Media. p. 536. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4471-5201-9" title="Special:BookSources/978-1-4471-5201-9"><bdi>978-1-4471-5201-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+Theory&rft.pages=536&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4471-5201-9&rft.au=Alexander+A.+Borovkov&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhRk_AAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-Yakir2013page5-328"><a href="#cite_ref-Yakir2013page5_328-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenjamin_Yakir2013" class="citation book cs1">Benjamin Yakir (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HShwAAAAQBAJ&pg=PT97">Extremes in Random Fields: A Theory and Its Applications</a>. John Wiley & Sons. p. 5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-72062-2" title="Special:BookSources/978-1-118-72062-2"><bdi>978-1-118-72062-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Extremes+in+Random+Fields%3A+A+Theory+and+Its+Applications&rft.pages=5&rft.pub=John+Wiley+%26+Sons&rft.date=2013&rft.isbn=978-1-118-72062-2&rft.au=Benjamin+Yakir&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHShwAAAAQBAJ%26pg%3DPT97&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-329"><a href="#cite_ref-329">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlackScholes1973" class="citation journal cs1">Black, Fischer; Scholes, Myron (1973). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1831029">"The Pricing of Options and Corporate Liabilities"</a>. Journal of Political Economy. 81 (3): 637–654. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F260062">10.1086/260062</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-3808">0022-3808</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1831029">1831029</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Political+Economy&rft.atitle=The+Pricing+of+Options+and+Corporate+Liabilities&rft.volume=81&rft.issue=3&rft.pages=637-654&rft.date=1973&rft.issn=0022-3808&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1831029%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1086%2F260062&rft.aulast=Black&rft.aufirst=Fischer&rft.au=Scholes%2C+Myron&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1831029&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-330"><a href="#cite_ref-330">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMerton2005" class="citation cs2">Merton, Robert C. (July 2005), <a rel="nofollow" class="external text" href="http://www.worldscientific.com/doi/abs/10.1142/9789812701022_0008">"Theory of rational option pricing"</a>, Theory of Valuation (2 ed.), WORLD SCIENTIFIC, pp. 229–288, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F9789812701022_0008">10.1142/9789812701022_0008</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-374-3" title="Special:BookSources/978-981-256-374-3"><bdi>978-981-256-374-3</bdi></a>, retrieved 2024-09-30</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+of+Valuation&rft.atitle=Theory+of+rational+option+pricing&rft.pages=229-288&rft.date=2005-07&rft_id=info%3Adoi%2F10.1142%2F9789812701022_0008&rft.isbn=978-981-256-374-3&rft.aulast=Merton&rft.aufirst=Robert+C.&rft_id=http%3A%2F%2Fwww.worldscientific.com%2Fdoi%2Fabs%2F10.1142%2F9789812701022_0008&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-331"><a href="#cite_ref-331">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeston1993" class="citation journal cs1">Heston, Steven L. (1993). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2962057">"A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options"</a>. The Review of Financial Studies. 6 (2): 327–343. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Frfs%2F6.2.327">10.1093/rfs/6.2.327</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0893-9454">0893-9454</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2962057">2962057</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Review+of+Financial+Studies&rft.atitle=A+Closed-Form+Solution+for+Options+with+Stochastic+Volatility+with+Applications+to+Bond+and+Currency+Options&rft.volume=6&rft.issue=2&rft.pages=327-343&rft.date=1993&rft.issn=0893-9454&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2962057%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1093%2Frfs%2F6.2.327&rft.aulast=Heston&rft.aufirst=Steven+L.&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2962057&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-332"><a href="#cite_ref-332">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoss2010" class="citation book cs1">Ross, Sheldon M. (2010). Introduction to probability models (10th ed.). Amsterdam Heidelberg: Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-375686-2" title="Special:BookSources/978-0-12-375686-2"><bdi>978-0-12-375686-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+probability+models&rft.place=Amsterdam+Heidelberg&rft.edition=10th&rft.pub=Elsevier&rft.date=2010&rft.isbn=978-0-12-375686-2&rft.aulast=Ross&rft.aufirst=Sheldon+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> <li id="cite_note-333"><a href="#cite_ref-333">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoss2010" class="citation book cs1">Ross, Sheldon M. (2010). Introduction to probability models (10th ed.). Amsterdam Heidelberg: Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-375686-2" title="Special:BookSources/978-0-12-375686-2"><bdi>978-0-12-375686-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+probability+models&rft.place=Amsterdam+Heidelberg&rft.edition=10th&rft.pub=Elsevier&rft.date=2010&rft.isbn=978-0-12-375686-2&rft.aulast=Ross&rft.aufirst=Sheldon+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=62" title="Edit section: Further reading">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Further_reading_cleanup plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This "<a href="/wiki/Wikipedia:Manual_of_Style/Layout#Further_reading" title="Wikipedia:Manual of Style/Layout">Further reading</a>" section may need cleanup. Please read the <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">editing guide</a> and help improve the section. (July 2018) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Articles">Articles</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=63" title="Edit section: Articles">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApplebaum2004" class="citation journal cs1">Applebaum, David (2004). "Lévy processes: From probability to finance and quantum groups". Notices of the AMS. 51 (11): 1336–1347.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+AMS&rft.atitle=L%C3%A9vy+processes%3A+From+probability+to+finance+and+quantum+groups&rft.volume=51&rft.issue=11&rft.pages=1336-1347&rft.date=2004&rft.aulast=Applebaum&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCramer1976" class="citation journal cs1">Cramer, Harald (1976). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faop%2F1176996025">"Half a Century with Probability Theory: Some Personal Recollections"</a>. The Annals of Probability. 4 (4): 509–546. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faop%2F1176996025">10.1214/aop/1176996025</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0091-1798">0091-1798</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Probability&rft.atitle=Half+a+Century+with+Probability+Theory%3A+Some+Personal+Recollections&rft.volume=4&rft.issue=4&rft.pages=509-546&rft.date=1976&rft_id=info%3Adoi%2F10.1214%2Faop%2F1176996025&rft.issn=0091-1798&rft.aulast=Cramer&rft.aufirst=Harald&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faop%252F1176996025&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuttorpThorarinsdottir2012" class="citation journal cs1">Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes". International Statistical Review. 80 (2): 253–268. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1751-5823.2012.00181.x">10.1111/j.1751-5823.2012.00181.x</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0306-7734">0306-7734</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:80836">80836</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Statistical+Review&rft.atitle=What+Happened+to+Discrete+Chaos%2C+the+Quenouille+Process%2C+and+the+Sharp+Markov+Property%3F+Some+History+of+Stochastic+Point+Processes&rft.volume=80&rft.issue=2&rft.pages=253-268&rft.date=2012&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A80836%23id-name%3DS2CID&rft.issn=0306-7734&rft_id=info%3Adoi%2F10.1111%2Fj.1751-5823.2012.00181.x&rft.aulast=Guttorp&rft.aufirst=Peter&rft.au=Thorarinsdottir%2C+Thordis+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJarrowProtter2004" class="citation book cs1">Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: the early years, 1880–1970". A Festschrift for Herman Rubin. Institute of Mathematical Statistics Lecture Notes - Monograph Series. pp. 75–91. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Flnms%2F1196285381">10.1214/lnms/1196285381</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-940600-61-4" title="Special:BookSources/978-0-940600-61-4"><bdi>978-0-940600-61-4</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0749-2170">0749-2170</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+short+history+of+stochastic+integration+and+mathematical+finance%3A+the+early+years%2C+1880%E2%80%931970&rft.btitle=A+Festschrift+for+Herman+Rubin&rft.series=Institute+of+Mathematical+Statistics+Lecture+Notes+-+Monograph+Series&rft.pages=75-91&rft.date=2004&rft.issn=0749-2170&rft_id=info%3Adoi%2F10.1214%2Flnms%2F1196285381&rft.isbn=978-0-940600-61-4&rft.aulast=Jarrow&rft.aufirst=Robert&rft.au=Protter%2C+Philip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeyer2009" class="citation journal cs1">Meyer, Paul-André (2009). "Stochastic Processes from 1950 to the Present". Electronic Journal for History of Probability and Statistics. 5 (1): 1–42.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Journal+for+History+of+Probability+and+Statistics&rft.atitle=Stochastic+Processes+from+1950+to+the+Present&rft.volume=5&rft.issue=1&rft.pages=1-42&rft.date=2009&rft.aulast=Meyer&rft.aufirst=Paul-Andr%C3%A9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Books">Books</h3>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=64" title="Edit section: Books">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_J._Adler2010" class="citation book cs1">Robert J. Adler (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ryejJmJAj28C&pg=PA263">The Geometry of Random Fields</a>. SIAM. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-693-1" title="Special:BookSources/978-0-89871-693-1"><bdi>978-0-89871-693-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Random+Fields&rft.pub=SIAM&rft.date=2010&rft.isbn=978-0-89871-693-1&rft.au=Robert+J.+Adler&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DryejJmJAj28C%26pg%3DPA263&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_J._AdlerJonathan_E._Taylor2009" class="citation book cs1">Robert J. Adler; Jonathan E. Taylor (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R5BGvQ3ejloC">Random Fields and Geometry</a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-48116-6" title="Special:BookSources/978-0-387-48116-6"><bdi>978-0-387-48116-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Fields+and+Geometry&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=978-0-387-48116-6&rft.au=Robert+J.+Adler&rft.au=Jonathan+E.+Taylor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DR5BGvQ3ejloC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPierre_Brémaud2013" class="citation book cs1">Pierre Brémaud (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jrPVBwAAQBAJ">Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues</a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-3124-8" title="Special:BookSources/978-1-4757-3124-8"><bdi>978-1-4757-3124-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Markov+Chains%3A+Gibbs+Fields%2C+Monte+Carlo+Simulation%2C+and+Queues&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013&rft.isbn=978-1-4757-3124-8&rft.au=Pierre+Br%C3%A9maud&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjrPVBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_L._Doob1990" class="citation book cs1"><a href="/wiki/Joseph_L._Doob" title="Joseph L. Doob">Joseph L. Doob</a> (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NrsrAAAAYAAJ">Stochastic processes</a>. Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+processes&rft.pub=Wiley&rft.date=1990&rft.au=Joseph+L.+Doob&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNrsrAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnders_Hald2005" class="citation book cs1">Anders Hald (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pOQy6-qnVx8C">A History of Probability and Statistics and Their Applications before 1750</a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-72517-6" title="Special:BookSources/978-0-471-72517-6"><bdi>978-0-471-72517-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Probability+and+Statistics+and+Their+Applications+before+1750&rft.pub=John+Wiley+%26+Sons&rft.date=2005&rft.isbn=978-0-471-72517-6&rft.au=Anders+Hald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpOQy6-qnVx8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrispin_Gardiner2010" class="citation book cs1"><a href="/wiki/Crispin_Gardiner" title="Crispin Gardiner">Crispin Gardiner</a> (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=321EuQAACAAJ&q=Stochastic+methods">Stochastic Methods</a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-70712-7" title="Special:BookSources/978-3-540-70712-7"><bdi>978-3-540-70712-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Methods&rft.pub=Springer&rft.date=2010&rft.isbn=978-3-540-70712-7&rft.au=Crispin+Gardiner&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D321EuQAACAAJ%26q%3DStochastic%2Bmethods&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIosif_I._GikhmanAnatoly_Vladimirovich_Skorokhod1996" class="citation book cs1">Iosif I. Gikhman; Anatoly Vladimirovich Skorokhod (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2">Introduction to the Theory of Random Processes</a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-69387-3" title="Special:BookSources/978-0-486-69387-3"><bdi>978-0-486-69387-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Random+Processes&rft.pub=Courier+Corporation&rft.date=1996&rft.isbn=978-0-486-69387-3&rft.au=Iosif+I.+Gikhman&rft.au=Anatoly+Vladimirovich+Skorokhod&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyJyLzG7N7r8C%26pg%3DPR2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmanuel_Parzen2015" class="citation book cs1"><a href="/wiki/Emanuel_Parzen" title="Emanuel Parzen">Emanuel Parzen</a> (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0mB2CQAAQBAJ">Stochastic Processes</a>. Courier Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-79688-8" title="Special:BookSources/978-0-486-79688-8"><bdi>978-0-486-79688-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+Processes&rft.pub=Courier+Dover+Publications&rft.date=2015&rft.isbn=978-0-486-79688-8&rft.au=Emanuel+Parzen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0mB2CQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurray_Rosenblatt1962" class="citation book cs1"><a href="/wiki/Murray_Rosenblatt" title="Murray Rosenblatt">Murray Rosenblatt</a> (1962). <a rel="nofollow" class="external text" href="https://archive.org/details/randomprocesses00rose_0">Random Processes</a>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Processes&rft.pub=Oxford+University+Press&rft.date=1962&rft.au=Murray+Rosenblatt&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Frandomprocesses00rose_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStochastic+process" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Stochastic_process&action=edit&section=65" title="Edit section: External links">edit</a>]</div> <ul><li><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Stochastic_processes" title="Template:Stochastic processes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Stochastic_processes" title="Template talk:Stochastic processes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Stochastic_processes" title="Special:EditPage/Template:Stochastic processes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Stochastic_processes" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li> <li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li> <li><a href="/wiki/Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li> <li><a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li> <li><a href="/wiki/Moran_process" title="Moran process">Moran process</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a> <ul><li><a href="/wiki/Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li> <li><a href="/wiki/Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li> <li><a href="/wiki/Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li> <li><a href="/wiki/Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_process" title="Additive process">Additive process</a></li> <li><a href="/wiki/Bessel_process" title="Bessel process">Bessel process</a></li> <li><a href="/wiki/Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a> <ul><li><a href="/wiki/Birth_process" title="Birth process">pure birth</a></li></ul></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a> <ul><li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Bridge</a></li> <li><a href="/wiki/Brownian_excursion" title="Brownian excursion">Excursion</a></li> <li><a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li> <li><a href="/wiki/Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li> <li><a href="/wiki/Brownian_meander" title="Brownian meander">Meander</a></li></ul></li> <li><a href="/wiki/Cauchy_process" title="Cauchy process">Cauchy process</a></li> <li><a href="/wiki/Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li> <li><a href="/wiki/Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li> <li><a href="/wiki/Cox_process" title="Cox process">Cox process</a></li> <li><a href="/wiki/Diffusion_process" title="Diffusion process">Diffusion process</a></li> <li><a href="/wiki/Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson Brownian motion</a></li> <li><a href="/wiki/Empirical_process" title="Empirical process">Empirical process</a></li> <li><a href="/wiki/Feller_process" title="Feller process">Feller process</a></li> <li><a href="/wiki/Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li> <li><a href="/wiki/Gamma_process" title="Gamma process">Gamma process</a></li> <li><a href="/wiki/Geometric_process" title="Geometric process">Geometric process</a></li> <li><a href="/wiki/Hawkes_process" title="Hawkes process">Hawkes process</a></li> <li><a href="/wiki/Hunt_process" title="Hunt process">Hunt process</a></li> <li><a href="/wiki/Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li> <li><a href="/wiki/It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li> <li><a href="/wiki/It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li> <li><a href="/wiki/Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li> <li><a href="/wiki/Jump_process" title="Jump process">Jump process</a></li> <li><a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a></li> <li><a href="/wiki/Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li> <li><a href="/wiki/Markov_additive_process" title="Markov additive process">Markov additive process</a></li> <li><a href="/wiki/McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li> <li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="Ornstein–Uhlenbeck process">Ornstein–Uhlenbeck process</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson process</a> <ul><li><a href="/wiki/Compound_Poisson_process" title="Compound Poisson process">Compound</a></li> <li><a href="/wiki/Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li> <li><a href="/wiki/Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li> <li><a href="/wiki/Semimartingale" title="Semimartingale">Semimartingale</a></li> <li><a href="/wiki/Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li> <li><a href="/wiki/Stable_process" title="Stable process">Stable process</a></li> <li><a href="/wiki/Superprocess" title="Superprocess">Superprocess</a></li> <li><a href="/wiki/Telegraph_process" title="Telegraph process">Telegraph process</a></li> <li><a href="/wiki/Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li> <li><a href="/wiki/Wiener_process" title="Wiener process">Wiener process</a></li> <li><a href="/wiki/Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branching_process" title="Branching process">Branching process</a></li> <li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian process</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li> <li><a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li> <li><a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a> <ul><li><a href="/wiki/Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li> <li><a href="/wiki/Local_martingale" title="Local martingale">Local</a></li> <li><a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">Sub-</a></li> <li><a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li> <li><a href="/wiki/Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li> <li><a href="/wiki/Regenerative_process" title="Regenerative process">Regenerative process</a></li> <li><a href="/wiki/Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li> <li><a href="/wiki/Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li> <li><a href="/wiki/Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li> <li><a href="/wiki/Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li> <li><a href="/wiki/Ising_model" title="Ising model">Ising model</a> <ul><li><a href="/wiki/Potts_model" title="Potts model">Potts model</a></li> <li><a href="/wiki/Boolean_network" title="Boolean network">Boolean network</a></li></ul></li> <li><a href="/wiki/Markov_random_field" title="Markov random field">Markov random field</a></li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation</a></li> <li><a href="/wiki/Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li> <li><a href="/wiki/Point_process" title="Point process">Point process</a> <ul><li><a href="/wiki/Point_process#Cox_point_process" title="Point process">Cox</a></li> <li><a href="/wiki/Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li> <li><a href="/wiki/Random_field" title="Random field">Random field</a></li> <li><a href="/wiki/Random_graph" title="Random graph">Random graph</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li> <li><a href="/wiki/Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li> <li><a href="/wiki/Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li> <li><a href="/wiki/Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li> <li><a href="/wiki/Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li> <li><a href="/wiki/Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li> <li><a href="/wiki/Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li> <li><a href="/wiki/Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li> <li><a href="/wiki/Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li> <li><a href="/wiki/Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li> <li><a href="/wiki/Chen_model" title="Chen model">Chen</a></li> <li><a href="/wiki/Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li> <li><a href="/wiki/Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li> <li><a href="/wiki/Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li> <li><a href="/wiki/Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li> <li><a href="/wiki/Heston_model" title="Heston model">Heston</a></li> <li><a href="/wiki/Ho%E2%80%93Lee_model" title="Ho–Lee model">Ho–Lee</a></li> <li><a href="/wiki/Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li> <li><a href="/wiki/Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li> <li><a href="/wiki/LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li> <li><a href="/wiki/Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li> <li><a href="/wiki/SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li> <li><a href="/wiki/Vasicek_model" title="Vasicek model">Vašíček</a></li> <li><a href="/wiki/Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li> <li><a href="/wiki/Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li> <li><a href="/wiki/Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li> <li><a href="/wiki/Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bulk_queue" title="Bulk queue">Bulk</a></li> <li><a href="/wiki/Fluid_queue" title="Fluid queue">Fluid</a></li> <li><a href="/wiki/G-network" title="G-network">Generalized queueing network</a></li> <li><a href="/wiki/M/G/1_queue" title="M/G/1 queue">M/G/1</a></li> <li><a href="/wiki/M/M/1_queue" title="M/M/1 queue">M/M/1</a></li> <li><a href="/wiki/M/M/c_queue" title="M/M/c queue">M/M/c</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li> <li><a href="/wiki/Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li> <li><a href="/wiki/Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li> <li><a href="/wiki/Ergodicity" title="Ergodicity">Ergodic</a></li> <li><a href="/wiki/Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li> <li><a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li> <li><a href="/wiki/Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li> <li><a href="/wiki/Markov_property" title="Markov property">Markov</a></li> <li><a href="/wiki/Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li> <li><a href="/wiki/Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li> <li><a href="/wiki/Predictable_process" title="Predictable process">Predictable</a></li> <li><a href="/wiki/Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li> <li><a href="/wiki/Self-similar_process" title="Self-similar process">Self-similar</a></li> <li><a href="/wiki/Stationary_process" title="Stationary process">Stationary</a></li> <li><a href="/wiki/Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li> <li><a href="/wiki/Donsker%27s_theorem" title="Donsker's theorem">Donsker's theorem</a></li> <li><a href="/wiki/Doob%27s_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li> <li><a href="/wiki/Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li> <li><a href="/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li> <li><a href="/wiki/Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li> <li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li> <li><a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li> <li><a href="/wiki/Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li> <li><a href="/wiki/Sanov%27s_theorem" title="Sanov's theorem">Sanov's theorem</a></li> <li><a href="/wiki/Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="/wiki/Blumenthal%27s_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="/wiki/Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="/wiki/Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="/wiki/Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="/wiki/Kolmogorov%27s_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="/wiki/L%C3%A9vy%27s_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li> <li><a href="/wiki/Doob%27s_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li> <li><a href="/wiki/Doob%27s_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li> <li><a href="/wiki/Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li> <li><a href="/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li> <li><a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li> <li><a href="/wiki/Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li> <li><a href="/wiki/Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li> <li><a href="/wiki/Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li> <li><a href="/wiki/Doob%27s_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li> <li><a href="/wiki/Dynkin%27s_formula" title="Dynkin's formula">Dynkin's formula</a></li> <li><a href="/wiki/Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li> <li><a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li> <li><a href="/wiki/Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li> <li><a href="/wiki/Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li> <li><a href="/wiki/It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li> <li><a href="/wiki/It%C3%B4%27s_lemma" title="Itô's lemma">Itô's lemma</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li> <li><a href="/wiki/Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li> <li><a href="/wiki/Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li> <li><a href="/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li> <li><a href="/wiki/Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li> <li><a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li> <li><a href="/wiki/Prokhorov%27s_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li> <li><a href="/wiki/Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li> <li><a href="/wiki/Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li> <li><a href="/wiki/Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li> <li><a href="/wiki/Skorokhod%27s_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li> <li><a href="/wiki/Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li> <li><a href="/wiki/Snell_envelope" title="Snell envelope">Snell envelope</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a> <ul><li><a href="/wiki/Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li> <li><a href="/wiki/Stopping_time" title="Stopping time">Stopping time</a></li> <li><a href="/wiki/Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li> <li><a href="/wiki/Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li> <li><a href="/wiki/Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li> <li><a href="/wiki/Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical</a></li> <li><a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li> <li><a href="/wiki/Stochastic_control" title="Stochastic control">Control theory</a></li> <li><a href="/wiki/Econometrics" title="Econometrics">Econometrics</a></li> <li><a href="/wiki/Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li> <li><a href="/wiki/Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li> <li><a href="/wiki/Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Queueing_theory" title="Queueing theory">Queueing theory</a></li> <li><a href="/wiki/Renewal_theory" title="Renewal theory">Renewal theory</a></li> <li><a href="/wiki/Ruin_theory" title="Ruin theory">Ruin theory</a></li> <li><a href="/wiki/Signal_processing" title="Signal processing">Signal processing</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li> <li><a href="/wiki/Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li> <li><a href="/wiki/Category:Stochastic_processes" title="Category:Stochastic processes">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" 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alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b7f745dd4-8mfqd","wgBackendResponseTime":167,"wgPageParseReport":{"limitreport":{"cputime":"2.868","walltime":"3.113","ppvisitednodes":{"value":20312,"limit":1000000},"postexpandincludesize":{"value":667561,"limit":2097152},"templateargumentsize":{"value":6693,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":14,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":1291067,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 2360.878 1 -total"," 70.03% 1653.356 2 Template:Reflist"," 47.06% 1111.147 272 Template:Cite_book"," 10.74% 253.566 57 Template:Cite_journal"," 5.45% 128.749 3 Template:Harvtxt"," 3.49% 82.314 1 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[\"CITEREFBert_E._FristedtLawrence_F._Gray2013\"] = 1,\n [\"CITEREFBertsekas1996\"] = 1,\n [\"CITEREFBertsekasTsitsiklis2002\"] = 1,\n [\"CITEREFBingham2000\"] = 1,\n [\"CITEREFBingham2005\"] = 1,\n [\"CITEREFBlackScholes1973\"] = 1,\n [\"CITEREFBressloff2014\"] = 1,\n [\"CITEREFBruHertz2001\"] = 1,\n [\"CITEREFBruce_Hajek2015\"] = 1,\n [\"CITEREFBrush1958\"] = 1,\n [\"CITEREFBrush1967\"] = 1,\n [\"CITEREFBrush1968\"] = 4,\n [\"CITEREFCharles_Miller_GrinsteadJames_Laurie_Snell1997\"] = 1,\n [\"CITEREFChung1998\"] = 1,\n [\"CITEREFCourtaultKabanovBruCrepel2000\"] = 1,\n [\"CITEREFCoxIsham1980\"] = 1,\n [\"CITEREFCramer1976\"] = 2,\n [\"CITEREFCramér1969\"] = 1,\n [\"CITEREFCrispin_Gardiner2010\"] = 1,\n [\"CITEREFD.J._DaleyD._Vere-Jones2006\"] = 6,\n [\"CITEREFD.J._DaleyDavid_Vere-Jones2007\"] = 1,\n [\"CITEREFDani_GamermanHedibert_F._Lopes2006\"] = 1,\n [\"CITEREFDaniel_RevuzMarc_Yor2013\"] = 3,\n [\"CITEREFDavar_Khoshnevisan2006\"] = 3,\n [\"CITEREFDavid1955\"] = 1,\n [\"CITEREFDavid_Applebaum2004\"] = 6,\n [\"CITEREFDavid_Stirzaker2005\"] = 1,\n [\"CITEREFDavid_Williams1991\"] = 3,\n [\"CITEREFDonald_L._SnyderMichael_I._Miller2012\"] = 5,\n [\"CITEREFDoob1934\"] = 1,\n [\"CITEREFDoob1996\"] = 1,\n [\"CITEREFDougherty1999\"] = 1,\n [\"CITEREFDynkin1989\"] = 1,\n [\"CITEREFEllis1995\"] = 1,\n [\"CITEREFEmanuel_Parzen2015\"] = 3,\n [\"CITEREFEmbrechtsFreyFurrer2001\"] = 1,\n [\"CITEREFFima_C._Klebaner2005\"] = 9,\n [\"CITEREFFlorescu2014\"] = 1,\n [\"CITEREFFrancois_BaccelliPierre_Bremaud2013\"] = 1,\n [\"CITEREFG._LatoucheV._Ramaswami1999\"] = 1,\n [\"CITEREFGeoffrey_GrimmettDavid_Stirzaker2001\"] = 2,\n [\"CITEREFGeorg_LindgrenHolger_RootzenMaria_Sandsten2013\"] = 1,\n [\"CITEREFGetoor2009\"] = 1,\n [\"CITEREFGilles_Pisier2016\"] = 1,\n [\"CITEREFGlassermanKou2006\"] = 1,\n [\"CITEREFGregory_F._LawlerVlada_Limic2010\"] = 2,\n [\"CITEREFGusakKukushKulikMishura2010\"] = 1,\n [\"CITEREFGuttorpThorarinsdottir2012\"] = 2,\n [\"CITEREFHald1981\"] = 1,\n [\"CITEREFHayes2013\"] = 1,\n [\"CITEREFHenk_C._Tijms2003\"] = 1,\n [\"CITEREFHeston1993\"] = 1,\n [\"CITEREFHiroshi_Kunita1997\"] = 1,\n [\"CITEREFIbe2013\"] = 1,\n [\"CITEREFIlya_Molchanov2005\"] = 1,\n [\"CITEREFIoannis_KaratzasSteven_Shreve1991\"] = 7,\n [\"CITEREFIonut_Florescu2014\"] = 10,\n [\"CITEREFIosif_I._GikhmanAnatoly_Vladimirovich_Skorokhod1969\"] = 1,\n [\"CITEREFIosif_I._GikhmanAnatoly_Vladimirovich_Skorokhod1996\"] = 1,\n [\"CITEREFIosif_Ilyich_GikhmanAnatoly_Vladimirovich_Skorokhod1969\"] = 5,\n [\"CITEREFJ._F._C._Kingman1992\"] = 5,\n [\"CITEREFJ._Michael_Steele2012\"] = 8,\n [\"CITEREFJarrowProtter2004\"] = 2,\n [\"CITEREFJean_Bertoin1998\"] = 2,\n [\"CITEREFJeffrey_S_Rosenthal2006\"] = 2,\n [\"CITEREFJesper_MollerRasmus_Plenge_Waagepetersen2003\"] = 1,\n [\"CITEREFJochen_BlathPeter_ImkellerSylvie_Roelly2011\"] = 1,\n [\"CITEREFJoel_Louis_Lebowitz1984\"] = 1,\n [\"CITEREFJohn_A._Gubner2006\"] = 1,\n [\"CITEREFJohn_Lamperti1977\"] = 4,\n [\"CITEREFJohn_Tabak2014\"] = 2,\n [\"CITEREFJoseph_L._Doob1990\"] = 5,\n [\"CITEREFJovanovic2012\"] = 1,\n [\"CITEREFKendallBatchelorBinghamHayman1990\"] = 2,\n [\"CITEREFKhintchine1934\"] = 1,\n [\"CITEREFKiyosi_Itō2006\"] = 2,\n [\"CITEREFKolmogoroff1931\"] = 1,\n [\"CITEREFKrishna_B._AthreyaSoumendra_N._Lahiri2006\"] = 3,\n [\"CITEREFL._C._G._RogersDavid_Williams2000\"] = 6,\n [\"CITEREFL._E._Maistrov2014\"] = 2,\n [\"CITEREFLaingLord2010\"] = 1,\n [\"CITEREFLandeEngenSæther2003\"] = 1,\n [\"CITEREFLauritzen1981\"] = 1,\n [\"CITEREFLeonard_Kleinrock1976\"] = 1,\n [\"CITEREFLeonid_KoralovYakov_G._Sinai2007\"] = 1,\n [\"CITEREFLindvall1991\"] = 1,\n [\"CITEREFLoïc_ChaumontMarc_Yor2012\"] = 1,\n [\"CITEREFM._Loève1978\"] = 1,\n [\"CITEREFMalte_HenkelDragi_Karevski2012\"] = 1,\n [\"CITEREFMarc_BarbutBernard_LockerLaurent_Mazliak2016\"] = 1,\n [\"CITEREFMark_A._PinskySamuel_Karlin2011\"] = 1,\n [\"CITEREFMartin_Haenggi2013\"] = 2,\n [\"CITEREFMerton2005\"] = 1,\n [\"CITEREFMeyer2009\"] = 2,\n [\"CITEREFMichel_Talagrand2014\"] = 1,\n [\"CITEREFMikhail_Lifshits2012\"] = 1,\n [\"CITEREFMonique_JeanblancMarc_YorMarc_Chesney2009\"] = 1,\n [\"CITEREFMurray_Rosenblatt1962\"] = 4,\n [\"CITEREFMusielaRutkowski2006\"] = 1,\n [\"CITEREFNicholas_H._BinghamRüdiger_Kiesel2013\"] = 1,\n [\"CITEREFO._B._Sheĭnin2006\"] = 1,\n [\"CITEREFOlav_Kallenberg2002\"] = 5,\n [\"CITEREFOliver_C._Ibe2013\"] = 1,\n [\"CITEREFOscar_SheyninHeinrich_Strecker2011\"] = 1,\n [\"CITEREFP._HallC._C._Heyde2014\"] = 3,\n [\"CITEREFPatrick_Billingsley2008\"] = 5,\n [\"CITEREFPatrick_Billingsley2013\"] = 1,\n [\"CITEREFPaulBaschnagel2013\"] = 1,\n [\"CITEREFPaul_C._Bressloff2014\"] = 1,\n [\"CITEREFPetar_Todorovic2012\"] = 2,\n [\"CITEREFPeter_E._KloedenEckhard_Platen2013\"] = 1,\n [\"CITEREFPeter_K._FrizNicolas_B._Victoir2010\"] = 1,\n [\"CITEREFPeter_MörtersYuval_Peres2010\"] = 3,\n [\"CITEREFPierre_Bremaud2013\"] = 2,\n [\"CITEREFPierre_Brémaud2013\"] = 1,\n [\"CITEREFPierre_Brémaud2014\"] = 3,\n [\"CITEREFQuastel2015\"] = 1,\n [\"CITEREFRaussenSkau2008\"] = 1,\n [\"CITEREFReuven_Y._RubinsteinDirk_P._Kroese2011\"] = 1,\n [\"CITEREFRichard_F._Bass2011\"] = 2,\n [\"CITEREFRichard_Serfozo2009\"] = 1,\n [\"CITEREFRick_Durrett2010\"] = 2,\n [\"CITEREFRobert_J._Adler2010\"] = 4,\n [\"CITEREFRobert_J._AdlerJonathan_E._Taylor2009\"] = 3,\n [\"CITEREFRoss2010\"] = 2,\n [\"CITEREFRoy_L._Streit2010\"] = 1,\n [\"CITEREFSamuel_KarlinHoward_E._Taylor2012\"] = 10,\n [\"CITEREFSean_MeynRichard_L._Tweedie2009\"] = 1,\n [\"CITEREFSeneta1998\"] = 1,\n [\"CITEREFSeneta2006\"] = 1,\n [\"CITEREFSheldon_M._Ross1996\"] = 3,\n [\"CITEREFShreve2004\"] = 1,\n [\"CITEREFSidney_I._Resnick2013\"] = 1,\n [\"CITEREFSnell2005\"] = 1,\n [\"CITEREFSteele2001\"] = 1,\n [\"CITEREFSteven_E._Shreve2004\"] = 1,\n [\"CITEREFStirzaker2000\"] = 1,\n [\"CITEREFSung_Nok_ChiuDietrich_StoyanWilfrid_S._KendallJoseph_Mecke2013\"] = 4,\n [\"CITEREFSøren_Asmussen2003\"] = 3,\n [\"CITEREFThiele1880\"] = 1,\n [\"CITEREFThomas_M._CoverJoy_A._Thomas2012\"] = 1,\n [\"CITEREFTruesdell1975\"] = 2,\n [\"CITEREFUbbo_F._Wiersema2008\"] = 1,\n [\"CITEREFValeriy_Skorokhod2005\"] = 4,\n [\"CITEREFVan_Kampen2011\"] = 1,\n [\"CITEREFVere-Jones2006\"] = 2,\n [\"CITEREFVladimir_I._Bogachev2007\"] = 1,\n [\"CITEREFVolker_Schmidt2014\"] = 1,\n [\"CITEREFWard_Whitt2006\"] = 3,\n [\"CITEREFWeiss2006\"] = 1,\n [\"CITEREFWilliam_J._Anderson2012\"] = 1,\n [\"CITEREFY.A._Rozanov2012\"] = 2,\n}\ntemplate_list = table#1 {\n [\"-\"] = 1,\n [\"Authority control\"] = 1,\n [\"Citation\"] = 1,\n [\"Cite OED\"] = 2,\n [\"Cite book\"] = 272,\n [\"Cite journal\"] = 57,\n [\"Color\"] = 2,\n [\"Columns-list\"] = 1,\n [\"Commons category-inline\"] = 1,\n [\"DEFAULTSORT:Stochastic Process\"] = 1,\n [\"Efn\"] = 10,\n [\"Further cleanup\"] = 1,\n [\"Harvtxt\"] = 3,\n [\"IPAc-en\"] = 1,\n [\"Industrial and applied mathematics\"] = 1,\n [\"Main\"] = 12,\n [\"Notelist\"] = 1,\n [\"Probability fundamentals\"] = 1,\n [\"Reflist\"] = 1,\n [\"Rp\"] = 4,\n [\"Short description\"] = 1,\n [\"Stochastic processes\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n","limitreport-profile":[["dataWrapper \u003Cmw.lua:672\u003E","220","14.3"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::callParserFunction","200","13.0"],["?","180","11.7"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::gsub","120","7.8"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::match","100","6.5"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getEntityStatements","60","3.9"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::setTTL","40","2.6"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::anchorEncode","40","2.6"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::preprocess","40","2.6"],["is_set 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