Additive process - Wikipedia

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class="vector-toc-numb">2</span> <span>Main properties</span> </div> </a> <button aria-controls="toc-Main_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Main properties subsection</span> </button> <ul id="toc-Main_properties-sublist" class="vector-toc-list"> <li id="toc-Independent_increments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Independent_increments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Independent increments</span> </div> </a> <ul id="toc-Independent_increments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuity_in_probability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuity_in_probability"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Continuity in probability</span> </div> </a> <ul id="toc-Continuity_in_probability-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lévy–Khintchine_representation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lévy–Khintchine_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lévy–Khintchine representation</span> </div> </a> <ul id="toc-Lévy–Khintchine_representation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Existence_and_uniqueness_in_law_of_additive_process" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Existence_and_uniqueness_in_law_of_additive_process"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Existence and uniqueness in law of additive process</span> </div> </a> <ul id="toc-Existence_and_uniqueness_in_law_of_additive_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subclass_of_additive_process" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Subclass_of_additive_process"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Subclass of additive process</span> </div> </a> <button aria-controls="toc-Subclass_of_additive_process-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Subclass of additive process subsection</span> </button> <ul id="toc-Subclass_of_additive_process-sublist" class="vector-toc-list"> <li id="toc-Additive_Logistic_Process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additive_Logistic_Process"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Additive Logistic Process</span> </div> </a> <ul id="toc-Additive_Logistic_Process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additive_Normal_Tempered_Stable_Process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additive_Normal_Tempered_Stable_Process"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Additive Normal Tempered Stable Process</span> </div> </a> <ul id="toc-Additive_Normal_Tempered_Stable_Process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additive_Subordinator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additive_Subordinator"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Additive Subordinator</span> </div> </a> <ul id="toc-Additive_Subordinator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sato_Process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sato_Process"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Sato 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id="toc-Characteristic_function_inversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characteristic_function_inversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Characteristic function inversion</span> </div> </a> <ul id="toc-Characteristic_function_inversion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Quantitative_finance" 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Cadlag in probability theory</div> <p>An <b>additive process</b>, in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, is a <a href="/wiki/Cadlag" class="mw-redirect" title="Cadlag">cadlag</a>, <a href="/wiki/Continuous_stochastic_process#Continuity_in_probability" title="Continuous stochastic process">continuous in probability</a> <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> with <a href="/wiki/Independent_increments" title="Independent increments">independent increments</a>. An additive process is the generalization of a <a href="/wiki/L%C3%A9vy_process" title="Lévy process">Lévy process</a> (a Lévy process is an additive process with stationary increments). An example of an additive process that is not a Lévy process is a <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> with a time-dependent drift.<sup id="cite_ref-FOOTNOTETankovCont2003455_1-0" class="reference"><a href="#cite_note-FOOTNOTETankovCont2003455-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The additive process was introduced by <a href="/wiki/Paul_L%C3%A9vy_(mathematician)" title="Paul Lévy (mathematician)">Paul Lévy</a> in 1937.<sup id="cite_ref-FOOTNOTETankovCont2003468_2-0" class="reference"><a href="#cite_note-FOOTNOTETankovCont2003468-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>There are applications of the additive process in <a href="/wiki/Quantitative_finance" class="mw-redirect" title="Quantitative finance">quantitative finance</a><sup id="cite_ref-FOOTNOTETankovCont2003454_3-0" class="reference"><a href="#cite_note-FOOTNOTETankovCont2003454-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> (this family of processes can capture important features of the <a href="/wiki/Implied_volatility" title="Implied volatility">implied volatility</a>) and in <a href="/wiki/Digital_image_processing" title="Digital image processing">digital image processing</a>.<sup id="cite_ref-FOOTNOTEBhattacharyaBrockwell197671_4-0" class="reference"><a href="#cite_note-FOOTNOTEBhattacharyaBrockwell197671-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An additive process is a generalization of a Lévy process obtained relaxing the hypothesis of stationary increments. Thanks to this feature an additive process can describe more complex phenomenons than a Lévy process. </p><p>A <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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 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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80025e1d695749922e65e153b3f33d0d94a75759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.002ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\geq 0}}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <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>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a341262173c747b2994e894e171b7c9b2a718c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.509ex;" alt="{\displaystyle X_{0}=0}"></span> almost surely is an additive process if it satisfy the following hypothesis: </p> <ol><li>It has independent increments.</li> <li>It is continuous in probability.<sup id="cite_ref-FOOTNOTETankovCont2003455_1-1" class="reference"><a href="#cite_note-FOOTNOTETankovCont2003455-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li></ol> <div class="mw-heading mw-heading2"><h2 id="Main_properties">Main properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=2" title="Edit section: Main properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Independent_increments">Independent increments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=3" title="Edit section: Independent increments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A stochastic process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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 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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80025e1d695749922e65e153b3f33d0d94a75759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.002ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\geq 0}}"></span> has independent increments if and only if for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq p<r\leq s<t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>r</mi> <mo>≤</mo> <mi>s</mi> <mo><</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq p<r\leq s<t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2efa1006585321ca5e8b3275fd8583d11a0ccdaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.704ex; height:2.509ex;" alt="{\displaystyle 0\leq p<r\leq s<t}"></span> the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}-X_{s}}"> <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> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}-X_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b215152f36de7f1363b8402b42cb4da30e10f02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.518ex; height:2.509ex;" alt="{\displaystyle X_{t}-X_{s}}"></span> is independent from the random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{r}-X_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{r}-X_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139d4d64af9b5b55be2a70b38a87f368b4c8c8c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.722ex; height:2.843ex;" alt="{\displaystyle X_{r}-X_{p}}"></span>.<sup id="cite_ref-FOOTNOTETankovCont200380_5-0" class="reference"><a href="#cite_note-FOOTNOTETankovCont200380-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This seems quite wrong. It is only pairwise independence. Actually, we need mutual independent for any finite number of disjoint intervals. This is well-described in the Wikipedia article on Levy processes. (August 2023)">clarification needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Continuity_in_probability">Continuity in probability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=4" title="Edit section: Continuity in probability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A stochastic process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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 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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80025e1d695749922e65e153b3f33d0d94a75759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.002ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\geq 0}}"></span> is continuous in probability if, and only if, for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29a2960e88369263fe3cfe00ccbfeb83daee212a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>0}"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{s\to t^{-}}\Pr \left({\big |}X_{s}-X_{t}{\big |}\geq \varepsilon \right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo stretchy="false">→</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <mo movablelimits="true" form="prefix">Pr</mo> <mrow> <mo>(</mo> <mrow> <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"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo>≥</mo> <mi>ε</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{s\to t^{-}}\Pr \left({\big |}X_{s}-X_{t}{\big |}\geq \varepsilon \right)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13002d6c51d1caeab274b7226e085acc3a57aea3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.51ex; height:4.509ex;" alt="{\displaystyle \lim _{s\to t^{-}}\Pr \left({\big |}X_{s}-X_{t}{\big |}\geq \varepsilon \right)=0.}"></span><sup id="cite_ref-FOOTNOTETankovCont200380_5-1" class="reference"><a href="#cite_note-FOOTNOTETankovCont200380-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Lévy–Khintchine_representation"><span id="L.C3.A9vy.E2.80.93Khintchine_representation"></span>Lévy–Khintchine representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=5" title="Edit section: Lévy–Khintchine representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a strong link between additive process and <a href="/wiki/Infinitely_divisible_distribution" class="mw-redirect" title="Infinitely divisible distribution">infinitely divisible distributions</a>. An additive process at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> has an infinitely divisible distribution characterized by the generating triplet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\gamma _{t},A_{t},\nu _{t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\gamma _{t},A_{t},\nu _{t})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3a1f0b37b79ec0a7bbf955663e643bbbf2a863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.451ex; height:2.843ex;" alt="{\displaystyle (\gamma _{t},A_{t},\nu _{t})}"></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c0cbac9d75722686c1079bbcb34f4d095dd2c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.03ex; height:2.176ex;" alt="{\displaystyle \gamma _{t}}"></span> is a vector in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <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>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265483c517cb98cde609f03a31964d86cdcb05c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.569ex; height:2.509ex;" alt="{\displaystyle A_{t}}"></span> is a matrix in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d\times d}}"> <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>d</mi> <mo>×</mo> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d\times d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a96e8d2e826900355660c8ce333f1ca2e08d07bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.908ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d\times d}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c91cfeb8214f212ac44b845696f658b381f44e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.974ex; height:2.009ex;" alt="{\displaystyle \nu _{t}}"></span> is a measure on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <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>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{t}(\{0\})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{t}(\{0\})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a683db6d81e84024c4651a26effd3262ed6449a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.532ex; height:2.843ex;" alt="{\displaystyle \nu _{t}(\{0\})=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{d}}(1\wedge x^{2})\nu _{t}(dx)<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>∧</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{d}}(1\wedge x^{2})\nu _{t}(dx)<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a785d15beeb848a9c01e66c0293830138e8dc003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.263ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{d}}(1\wedge x^{2})\nu _{t}(dx)<\infty }"></span>. <sup id="cite_ref-FOOTNOTESato199947_6-0" class="reference"><a href="#cite_note-FOOTNOTESato199947-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0c0cbac9d75722686c1079bbcb34f4d095dd2c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.03ex; height:2.176ex;" alt="{\displaystyle \gamma _{t}}"></span> is called drift term, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265483c517cb98cde609f03a31964d86cdcb05c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.569ex; height:2.509ex;" alt="{\displaystyle A_{t}}"></span> covariance matrix and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c91cfeb8214f212ac44b845696f658b381f44e35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.974ex; height:2.009ex;" alt="{\displaystyle \nu _{t}}"></span> Lévy measure. It is possible to write explicitly the additive process characteristic function using the <i>Lévy–Khintchine formula</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{X}(u)(t):=\operatorname {E} \left[e^{iu'X_{t}}\right]=\exp \left(u'\gamma _{t}i-{\frac {1}{2}}u'A_{t}u+\int _{\mathbb {R} ^{d}}\left(e^{iu'x}-1-iu'x\mathbf {I} _{|x|<1}\right)\,\nu _{t}(dx)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>i</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{X}(u)(t):=\operatorname {E} \left[e^{iu'X_{t}}\right]=\exp \left(u'\gamma _{t}i-{\frac {1}{2}}u'A_{t}u+\int _{\mathbb {R} ^{d}}\left(e^{iu'x}-1-iu'x\mathbf {I} _{|x|<1}\right)\,\nu _{t}(dx)\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfa0c2a46b91b3e03d54b7ed822bccbb911ce73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:85.31ex; height:6.176ex;" alt="{\displaystyle \varphi _{X}(u)(t):=\operatorname {E} \left[e^{iu'X_{t}}\right]=\exp \left(u'\gamma _{t}i-{\frac {1}{2}}u'A_{t}u+\int _{\mathbb {R} ^{d}}\left(e^{iu'x}-1-iu'x\mathbf {I} _{|x|<1}\right)\,\nu _{t}(dx)\right),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></span> is a vector in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <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>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I_{C}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I_{C}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.509ex;" alt="{\displaystyle \mathbf {I_{C}} }"></span> is the indicator function of the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>.<sup id="cite_ref-FOOTNOTESato199937–38_7-0" class="reference"><a href="#cite_note-FOOTNOTESato199937–38-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>A Lèvy process characteristic function has the same structure but with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{t}=t\gamma ,\nu _{t}=t\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>t</mi> <mi>γ</mi> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>t</mi> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{t}=t\gamma ,\nu _{t}=t\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f049c0e7f9cd1d9c74e438ef24fb115b3be7dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.409ex; height:2.509ex;" alt="{\displaystyle \gamma _{t}=t\gamma ,\nu _{t}=t\nu }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{t}=At}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{t}=At}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18df6ecec3a26f28a7be9ccf60aaa97af4cc9516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.25ex; height:2.509ex;" alt="{\displaystyle A_{t}=At}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> a vector in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <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>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> a positive definite matrix in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d\times d}}"> <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>d</mi> <mo>×</mo> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d\times d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a96e8d2e826900355660c8ce333f1ca2e08d07bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.908ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d\times d}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is a measure on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <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>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span>.<sup id="cite_ref-FOOTNOTETankovCont200395_8-0" class="reference"><a href="#cite_note-FOOTNOTETankovCont200395-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Existence_and_uniqueness_in_law_of_additive_process">Existence and uniqueness in law of additive process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=6" title="Edit section: Existence and uniqueness in law of additive process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following result together with the <i>Lévy–Khintchine formula</i> characterizes the additive process. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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 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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80025e1d695749922e65e153b3f33d0d94a75759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.002ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\geq 0}}"></span> be an additive process on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}"> <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>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a713426956296f1668fce772df3c60b9dde8a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}"></span>. Then, its infinitely divisible distribution is such that: </p> <ol><li>For all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265483c517cb98cde609f03a31964d86cdcb05c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.569ex; height:2.509ex;" alt="{\displaystyle A_{t}}"></span> is a positive definite matrix.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{0}=0,A_{0}=0,\nu _{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{0}=0,A_{0}=0,\nu _{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13158fa0f96b1531df2b8e6a3671b31638fe5a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.109ex; height:2.676ex;" alt="{\displaystyle \gamma _{0}=0,A_{0}=0,\nu _{0}=0}"></span> and for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,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,t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ce81dd4a9013d5a5502724585368c58ace5971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.964ex; height:2.343ex;" alt="{\displaystyle s,t}"></span> is such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s<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<t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2def280f2922bece4991a9bd1e06d83d093154" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.029ex; height:2.009ex;" alt="{\displaystyle s<t}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{t}-A_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{t}-A_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae2efdd3473263964b3c743b280cd0cf7d59bd4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.156ex; height:2.509ex;" alt="{\displaystyle A_{t}-A_{s}}"></span> is a positive definite matrix and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{t}(B)\geq \nu _{s}(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{t}(B)\geq \nu _{s}(B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c918695ebdb7b1ba1f63df6c198c2b7a68cd4400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.371ex; height:2.843ex;" alt="{\displaystyle \nu _{t}(B)\geq \nu _{s}(B)}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} (\mathbb {R} ^{d})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} (\mathbb {R} ^{d})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe90ac1e6d1f9d82dd46752da54fe9688a0b7bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.48ex; height:3.176ex;" alt="{\displaystyle \mathbf {B} (\mathbb {R} ^{d})}"></span>.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\to t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">→</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\to t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6acf6f2ef7df3150305fb3625ded360e546265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.544ex; height:2.009ex;" alt="{\displaystyle s\to t}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{s}\to \gamma _{t},A_{s}\to A_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{s}\to \gamma _{t},A_{s}\to A_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d12598394a23dc8893cfca13a6f439990c4acb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.815ex; height:2.676ex;" alt="{\displaystyle \gamma _{s}\to \gamma _{t},A_{s}\to A_{t}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{s}(B)\to \nu _{t}(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{s}(B)\to \nu _{t}(B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f5da974dfc9592a81b47d2859024d0baa28e45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.887ex; height:2.843ex;" alt="{\displaystyle \nu _{s}(B)\to \nu _{t}(B)}"></span> every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} (\mathbb {R} ^{d})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} (\mathbb {R} ^{d})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe90ac1e6d1f9d82dd46752da54fe9688a0b7bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.48ex; height:3.176ex;" alt="{\displaystyle \mathbf {B} (\mathbb {R} ^{d})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\not \in B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>∉</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\not \in B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d35d7ace11930313fb410c74f0401ee94b57a59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.767ex; height:2.676ex;" alt="{\displaystyle 0\not \in B}"></span>.</li></ol> <p>Conversely for family of infinitely divisible distributions characterized by a generating triplet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\gamma _{t},A_{t},\nu _{t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\gamma _{t},A_{t},\nu _{t})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3a1f0b37b79ec0a7bbf955663e643bbbf2a863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.451ex; height:2.843ex;" alt="{\displaystyle (\gamma _{t},A_{t},\nu _{t})}"></span> that satisfies 1, 2 and 3, it exists an additive process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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 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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80025e1d695749922e65e153b3f33d0d94a75759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.002ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\geq 0}}"></span> with this distribution.<sup id="cite_ref-FOOTNOTETankovCont2003458_9-0" class="reference"><a href="#cite_note-FOOTNOTETankovCont2003458-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTESato199963_10-0" class="reference"><a href="#cite_note-FOOTNOTESato199963-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Subclass_of_additive_process">Subclass of additive process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=7" title="Edit section: Subclass of additive process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Additive_Logistic_Process">Additive Logistic Process</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=8" title="Edit section: Additive Logistic Process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Family of additive processes with <a href="/wiki/Generalized_logistic_distribution" title="Generalized logistic distribution">generalized logistic distribution</a>. Their 5 parameters characteristic function is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[e^{iuX_{t}}\right]=\left({\frac {B(\alpha _{t}+i\sigma _{t}u,\beta _{t}-i\sigma u)}{B(\alpha _{t},\beta _{t})}}\right)^{\delta _{t}}e^{i\mu _{t}u}\;\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>u</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <mi>σ</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>u</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[e^{iuX_{t}}\right]=\left({\frac {B(\alpha _{t}+i\sigma _{t}u,\beta _{t}-i\sigma u)}{B(\alpha _{t},\beta _{t})}}\right)^{\delta _{t}}e^{i\mu _{t}u}\;\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/487a92cdf4e9aad826cf09ffee211cc51df23a96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.205ex; height:7.009ex;" alt="{\displaystyle \operatorname {E} \left[e^{iuX_{t}}\right]=\left({\frac {B(\alpha _{t}+i\sigma _{t}u,\beta _{t}-i\sigma u)}{B(\alpha _{t},\beta _{t})}}\right)^{\delta _{t}}e^{i\mu _{t}u}\;\;.}"></span></dd></dl> <p>Two subcases of additive logistic process are the symmetric logistic additive process with standard <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic distribution</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{t}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{t}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6be8b0004058b07e1a81a9d4840948844ccddc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.574ex; height:2.509ex;" alt="{\displaystyle \alpha _{t}=1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{t}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{t}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a375649e38d8a45097ea0965b01b302b561c03d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.403ex; height:2.509ex;" alt="{\displaystyle \beta _{t}=1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{t}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{t}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7622d2bb7d4dc2ad20ff2fb685ab7f82411c07a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.119ex; height:2.676ex;" alt="{\displaystyle \delta _{t}=1}"></span>) and the conjugate-power Dagum additive process with <a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum distribution</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{t}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{t}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6be8b0004058b07e1a81a9d4840948844ccddc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.574ex; height:2.509ex;" alt="{\displaystyle \alpha _{t}=1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{t}=1-\sigma (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{t}=1-\sigma (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db6f4715085a8ce26320a560177f8f7dab90680a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.222ex; height:2.843ex;" alt="{\displaystyle \beta _{t}=1-\sigma (t)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{t}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{t}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6be8b0004058b07e1a81a9d4840948844ccddc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.574ex; height:2.509ex;" alt="{\displaystyle \alpha _{t}=1}"></span>). </p><p>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8247e7beb0b3fa21c4fe35defef3a3cf93df67d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.228ex; height:2.176ex;" alt="{\displaystyle \mu _{t}}"></span> can always be chosen s.t. the additive process is a <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a>.<sup id="cite_ref-FOOTNOTECarrTorricelli2021698_11-0" class="reference"><a href="#cite_note-FOOTNOTECarrTorricelli2021698-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Additive_Normal_Tempered_Stable_Process">Additive Normal Tempered Stable Process</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=9" title="Edit section: Additive Normal Tempered Stable Process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Extension of the Lévy normal tempered stable processes; some well-known Lévy normal tempered stable processes have <a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">normal-inverse Gaussian distribution</a> and the <a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">variance-gamma distribution</a>. Additive normal tempered stable processes<sup id="cite_ref-FOOTNOTEAzzoneBaviera2022503_12-0" class="reference"><a href="#cite_note-FOOTNOTEAzzoneBaviera2022503-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> have the same characteristic function of Lévy normal tempered stable processes but with time dependent parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dd6db0238ac32f34c6feb604748e253e842356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle \sigma _{t}}"></span> (the level of the volatility), <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/352386366dcb1dcc184d82669513975561f2d2ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.037ex; height:2.509ex;" alt="{\displaystyle k_{t}}"></span> (the variance of jumps) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c5c704deeb3d13eff0896f46d975a51e0352228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.981ex; height:2.176ex;" alt="{\displaystyle \eta _{t}}"></span> (linked to the skew): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[e^{iuX_{t}}\right]={\cal {L}}_{t}\left(iu\left({\frac {1}{2}}+\eta _{t}\right)\sigma _{t}^{2}+{\frac {u^{2}\sigma _{t}^{2}}{2}};\;k_{t},\;\alpha \right)e^{iu\varphi _{t}t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>u</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>;</mo> <mspace width="thickmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>u</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[e^{iuX_{t}}\right]={\cal {L}}_{t}\left(iu\left({\frac {1}{2}}+\eta _{t}\right)\sigma _{t}^{2}+{\frac {u^{2}\sigma _{t}^{2}}{2}};\;k_{t},\;\alpha \right)e^{iu\varphi _{t}t},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7997c1a10aafde76e0e5348ae8730ea77226f414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.982ex; height:7.509ex;" alt="{\displaystyle \operatorname {E} \left[e^{iuX_{t}}\right]={\cal {L}}_{t}\left(iu\left({\frac {1}{2}}+\eta _{t}\right)\sigma _{t}^{2}+{\frac {u^{2}\sigma _{t}^{2}}{2}};\;k_{t},\;\alpha \right)e^{iu\varphi _{t}t},}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\cal {L}}_{t}\left(u;\;k_{t},\;\alpha \right):={\begin{cases}\displaystyle {\frac {t}{k_{t}}}\displaystyle {\frac {1-\alpha }{\alpha }}\left\{1-\left(1+{\frac {u\;k_{t}}{1-\alpha }}\right)^{\alpha }\right\}&{\mbox{if }}\;0<\alpha <1\\[4mm]\displaystyle -{\frac {t}{k_{t}}}\ln \left(1+u\;k_{t}\right)&{\mbox{if }}\;\alpha =0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>;</mo> <mspace width="thickmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="1.334em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mi>α</mi> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mspace width="thickmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mo>}</mo> </mrow> </mstyle> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mspace width="thickmathspace" /> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>u</mi> <mspace width="thickmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>if </mtext> </mstyle> </mrow> <mspace width="thickmathspace" /> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln {\cal {L}}_{t}\left(u;\;k_{t},\;\alpha \right):={\begin{cases}\displaystyle {\frac {t}{k_{t}}}\displaystyle {\frac {1-\alpha }{\alpha }}\left\{1-\left(1+{\frac {u\;k_{t}}{1-\alpha }}\right)^{\alpha }\right\}&{\mbox{if }}\;0<\alpha <1\\[4mm]\displaystyle -{\frac {t}{k_{t}}}\ln \left(1+u\;k_{t}\right)&{\mbox{if }}\;\alpha =0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fb9107f88a813a9041099ff378120238b98acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.331ex; margin-bottom: -0.174ex; width:68.755ex; height:14.176ex;" alt="{\displaystyle \ln {\cal {L}}_{t}\left(u;\;k_{t},\;\alpha \right):={\begin{cases}\displaystyle {\frac {t}{k_{t}}}\displaystyle {\frac {1-\alpha }{\alpha }}\left\{1-\left(1+{\frac {u\;k_{t}}{1-\alpha }}\right)^{\alpha }\right\}&{\mbox{if }}\;0<\alpha <1\\[4mm]\displaystyle -{\frac {t}{k_{t}}}\ln \left(1+u\;k_{t}\right)&{\mbox{if }}\;\alpha =0\end{cases}}}"></span></dd></dl> <p>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70d00573df93655cb061670a9638c70783d98b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.346ex; height:2.176ex;" alt="{\displaystyle \varphi _{t}}"></span> can always be chosen s.t. the additive process is a <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a>.<sup id="cite_ref-FOOTNOTEAzzoneBaviera2022503_12-1" class="reference"><a href="#cite_note-FOOTNOTEAzzoneBaviera2022503-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Additive_Subordinator">Additive Subordinator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=10" title="Edit section: Additive Subordinator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A positive non decreasing additive process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{S_{t}\}_{t\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>S</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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{S_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a7bc46477913d63df5fa4cd09597c45844ce9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.502ex; height:2.843ex;" alt="{\displaystyle \{S_{t}\}_{t\geq 0}}"></span> with values in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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 \mathbb {R} }"></span> is an additive <a href="/wiki/Subordinator_(mathematics)" title="Subordinator (mathematics)">subordinator</a>. An additive subordinator is a <a href="/wiki/Semimartingale" title="Semimartingale">semimartingale</a> (thanks to the fact that it is not decreasing) and it is always possible to rewrite its <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[e^{-uS_{t}}\right]=\exp \left(ub_{t}+\int _{\mathbb {R} ^{d}}(e^{iux}-1)\nu _{t}(dx)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>u</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>u</mi> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>d</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[e^{-uS_{t}}\right]=\exp \left(ub_{t}+\int _{\mathbb {R} ^{d}}(e^{iux}-1)\nu _{t}(dx)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/146ea58bf8ac5fea4a9354e873a3f7332e014cd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.902ex; height:6.176ex;" alt="{\displaystyle \operatorname {E} \left[e^{-uS_{t}}\right]=\exp \left(ub_{t}+\int _{\mathbb {R} ^{d}}(e^{iux}-1)\nu _{t}(dx)\right).}"></span><sup id="cite_ref-FOOTNOTELiLiMendoza-Arriaga20165–6_13-0" class="reference"><a href="#cite_note-FOOTNOTELiLiMendoza-Arriaga20165–6-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></dd></dl> <p>It is possible to use additive subordinator to time-change a Lévy process obtaining a new class of additive processes.<sup id="cite_ref-FOOTNOTELiLiMendoza-Arriaga20161_14-0" class="reference"><a href="#cite_note-FOOTNOTELiLiMendoza-Arriaga20161-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Sato_Process">Sato Process</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=11" title="Edit section: Sato Process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An additive <a href="/wiki/Self-similar_process" title="Self-similar process">self-similar process</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{Z_{t}\}_{t\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Z</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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{Z_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/053e50935f4a73a07c68d2624f310140ef23b375" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.665ex; height:2.843ex;" alt="{\displaystyle \{Z_{t}\}_{t\geq 0}}"></span> is called Sato process.<sup id="cite_ref-FOOTNOTEEberleinMadan20095_15-0" class="reference"><a href="#cite_note-FOOTNOTEEberleinMadan20095-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> It is possible to construct a Sato process from a Lévy process <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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 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> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{X_{t}\}_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80025e1d695749922e65e153b3f33d0d94a75759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.002ex; height:2.843ex;" alt="{\displaystyle \{X_{t}\}_{t\geq 0}}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4494df739b00ddebb3f741b5a5aab7415a78736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.413ex; height:2.509ex;" alt="{\displaystyle Z_{t}}"></span> has the same law of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{h}X_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{h}X_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8507832f4cbfada75b22e96c310a9e4d49ec7f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.997ex; height:3.009ex;" alt="{\displaystyle t^{h}X_{1}}"></span>. </p><p>An example is the variance gamma SSD, the Sato process obtained starting from the <a href="/wiki/Variance_gamma_process" title="Variance gamma process">variance gamma process</a>. </p><p>The characteristic function of the Variance gamma at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970dea4a5f5ec5355c4cdd62f6396fbc8b1baaa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=1}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[e^{iuX_{1}}\right]=\left({\frac {1}{1-iu\theta \nu +0.5\sigma ^{2}\nu u^{2}}}\right)^{1/\nu },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>u</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>u</mi> <mi>θ</mi> <mi>ν</mi> <mo>+</mo> <mn>0.5</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ν</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ν</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[e^{iuX_{1}}\right]=\left({\frac {1}{1-iu\theta \nu +0.5\sigma ^{2}\nu u^{2}}}\right)^{1/\nu },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a39c65c27e3a00b982531d989bc74c7471d8d8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.558ex; height:6.676ex;" alt="{\displaystyle \operatorname {E} \left[e^{iuX_{1}}\right]=\left({\frac {1}{1-iu\theta \nu +0.5\sigma ^{2}\nu u^{2}}}\right)^{1/\nu },}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ,\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>,</mo> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ,\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5e7cccf95bf9408ec76d52a9729d4a54e22863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.357ex; height:2.509ex;" alt="{\displaystyle \theta ,\nu }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> are positive constant. </p><p>The characteristic function of the variance gamma SSD is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[e^{iuZ_{t}}\right]=\left({\frac {1}{1-iut^{h}\theta \nu +0.5\sigma ^{2}\nu u^{2}t^{2}h}}\right)^{1/\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>u</mi> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mi>u</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msup> <mi>θ</mi> <mi>ν</mi> <mo>+</mo> <mn>0.5</mn> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ν</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[e^{iuZ_{t}}\right]=\left({\frac {1}{1-iut^{h}\theta \nu +0.5\sigma ^{2}\nu u^{2}t^{2}h}}\right)^{1/\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca44b33a9713a6e5ddb634c655aa53950d476add" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.74ex; height:6.676ex;" alt="{\displaystyle \operatorname {E} \left[e^{iuZ_{t}}\right]=\left({\frac {1}{1-iut^{h}\theta \nu +0.5\sigma ^{2}\nu u^{2}t^{2}h}}\right)^{1/\nu }}"></span><sup id="cite_ref-FOOTNOTECarrGemanMadanYor200739_16-0" class="reference"><a href="#cite_note-FOOTNOTECarrGemanMadanYor200739-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Simulation">Simulation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=12" title="Edit section: Simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Simulation of Additive process is computationally efficient thanks to the independence of increments. The additive process increments can be simulated separately and simulation can also be parallelized.<sup id="cite_ref-FOOTNOTEEberleinMadan2009_17-0" class="reference"><a href="#cite_note-FOOTNOTEEberleinMadan2009-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Jump_simulation">Jump simulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=13" title="Edit section: Jump simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p> Jump simulation is a generalization to the class of additive processes of the jump simulation technique developed for Lévy processes. The method is based on truncating small jumps below a certain threshold and simulating the finite number of independent jumps. Moreover, Gaussian approximation can be applied to replace small jumps with a diffusive term. It is also possible to use the <a href="/wiki/Ziggurat_algorithm" title="Ziggurat algorithm">Ziggurat algorithm</a> to speed up the simulation of jumps.<sup id="cite_ref-FOOTNOTEEberleinMadan200919_18-0" class="reference"><a href="#cite_note-FOOTNOTEEberleinMadan200919-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Characteristic_function_inversion">Characteristic function inversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=14" title="Edit section: Characteristic function inversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Simulation of Lévy process via characteristic function inversion is a well established technique in the literature.<sup id="cite_ref-FOOTNOTEBallottaKyriakou20141_19-0" class="reference"><a href="#cite_note-FOOTNOTEBallottaKyriakou20141-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> This technique can be extended to additive processes. The key idea is obtaining an approximation of the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF) by inverting the <a href="/wiki/Characteristic_function" title="Characteristic function">characteristic function</a>. The inversion speed is enhanced by the use of the <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">Fast Fourier transform</a>. Once the approximation of the CDF is available is it possible to simulate an additive process increment just by simulating a uniform random variable. The method has similar computational cost as simulating a standard geometric Brownian motion.<sup id="cite_ref-FOOTNOTEAzzoneBaviera20231–5_20-0" class="reference"><a href="#cite_note-FOOTNOTEAzzoneBaviera20231–5-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=15" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Quantitative_finance">Quantitative finance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=16" title="Edit section: Quantitative finance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lévy process is used to model the log-returns of market prices. Unfortunately, the <i>stationarity </i> of the increments does not reproduce correctly market data. A Lévy process fit well <a href="/wiki/Call_option" title="Call option">call option</a> and <a href="/wiki/Put_option" title="Put option">put option</a> prices (<a href="/wiki/Volatility_smile" title="Volatility smile">implied volatility</a>) for a single expiration date but is unable to fit options prices with different maturities (<a href="/wiki/Volatility_(finance)#Implied_volatility_parametrisation" title="Volatility (finance)">volatility surface</a>). The additive process introduces a <i>deterministic</i> non-stationarity that allows it to fit all expiration dates.<sup id="cite_ref-FOOTNOTETankovCont2003454_3-1" class="reference"><a href="#cite_note-FOOTNOTETankovCont2003454-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>A four-parameters Sato process (self-similar additive process) can reproduce correctly the volatility surface (3% error on the <a href="/wiki/S%26P_500_Index" class="mw-redirect" title="S&P 500 Index">S&P 500</a> equity market). This order of magnitude of error is usually obtained using models with 6-10 parameters to fit market data.<sup id="cite_ref-FOOTNOTECarrGemanMadanYor200732_21-0" class="reference"><a href="#cite_note-FOOTNOTECarrGemanMadanYor200732-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> A self-similar process correctly describes market data because of its flat <a href="/wiki/Skewness" title="Skewness">skewness</a> and excess <a href="/wiki/Kurtosis" title="Kurtosis">kurtosis</a>; empirical studies had observed this behavior in market skewness and excess kurtosis.<sup id="cite_ref-FOOTNOTECarrGemanMadanYor200737_22-0" class="reference"><a href="#cite_note-FOOTNOTECarrGemanMadanYor200737-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Some of the processes that fit option prices with a 3% error are VGSSD, NIGSSD, MXNRSSD obtained from variance gamma process, normal inverse Gaussian process and Meixner process.<sup id="cite_ref-FOOTNOTECarrGemanMadanYor200739–42_23-0" class="reference"><a href="#cite_note-FOOTNOTECarrGemanMadanYor200739–42-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>Additive normal tempered stable processes fit accurately equity market data ( error below 0.8% on the <a href="/wiki/S%26P_500_Index" class="mw-redirect" title="S&P 500 Index">S&P 500</a> equity market) specifically for short maturities. These family of processes reproduces very well also the equity market implied volatility skew. Moreover, an interesting power scaling characteristic arises in calibrated parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{t}={\bar {k}}t^{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{t}={\bar {k}}t^{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291519525d5e87b1eb8ad0cd947b5db5f241bdfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.361ex; height:3.009ex;" alt="{\displaystyle k_{t}={\bar {k}}t^{\beta }}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{t}={\bar {\eta }}t^{\delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>η</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{t}={\bar {\eta }}t^{\delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2cceaf6ab3fa2a157a23fb6da4079678ec8da7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.197ex; height:3.176ex;" alt="{\displaystyle \eta _{t}={\bar {\eta }}t^{\delta }}"></span>. There is statistical evidence that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73416922785589e358ae2bb10c7633667b4c24a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta =1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =-1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta =-1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3d870fbb1345357af23b6cd7720dec91b906b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.443ex; height:2.843ex;" alt="{\displaystyle \delta =-1/2}"></span>.<sup id="cite_ref-FOOTNOTEAzzoneBaviera2022506–508_24-0" class="reference"><a href="#cite_note-FOOTNOTEAzzoneBaviera2022506–508-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Lévy subordination is used to construct new Lévy processes (for example variance gamma process and normal inverse Gaussian process). There is a large number of financial applications of processes constructed by Lévy subordination. An additive process built via additive subordination maintains the analytical tractability of a process built via Lévy subordination but it reflects better the time-inhomogeneus structure of market data.<sup id="cite_ref-FOOTNOTELiLiMendoza-Arriaga20163_25-0" class="reference"><a href="#cite_note-FOOTNOTELiLiMendoza-Arriaga20163-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Additive subordination is applied to the commodity market<sup id="cite_ref-FOOTNOTELiLiMendoza-Arriaga201617_26-0" class="reference"><a href="#cite_note-FOOTNOTELiLiMendoza-Arriaga201617-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and to VIX options.<sup id="cite_ref-FOOTNOTELiLiZhang20171_27-0" class="reference"><a href="#cite_note-FOOTNOTELiLiZhang20171-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Digital_image_processing">Digital image processing</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=17" title="Edit section: Digital image processing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An estimator based on the minimum of an additive process can be applied to image processing. Such estimator aims to distinguish between real signal and noise in the picture pixels.<sup id="cite_ref-FOOTNOTEBhattacharyaBrockwell197671_4-1" class="reference"><a href="#cite_note-FOOTNOTEBhattacharyaBrockwell197671-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTETankovCont2003455-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTETankovCont2003455_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTETankovCont2003455_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFTankovCont2003">Tankov & Cont 2003</a>, p. 455.</span> </li> <li id="cite_note-FOOTNOTETankovCont2003468-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETankovCont2003468_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTankovCont2003">Tankov & Cont 2003</a>, p. 468.</span> </li> <li id="cite_note-FOOTNOTETankovCont2003454-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTETankovCont2003454_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTETankovCont2003454_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFTankovCont2003">Tankov & Cont 2003</a>, p. 454.</span> </li> <li id="cite_note-FOOTNOTEBhattacharyaBrockwell197671-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEBhattacharyaBrockwell197671_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEBhattacharyaBrockwell197671_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBhattacharyaBrockwell1976">Bhattacharya & Brockwell 1976</a>, p. 71.</span> </li> <li id="cite_note-FOOTNOTETankovCont200380-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTETankovCont200380_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTETankovCont200380_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFTankovCont2003">Tankov & Cont 2003</a>, p. 80.</span> </li> <li id="cite_note-FOOTNOTESato199947-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESato199947_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSato1999">Sato 1999</a>, p. 47.</span> </li> <li id="cite_note-FOOTNOTESato199937–38-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESato199937–38_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSato1999">Sato 1999</a>, pp. 37–38.</span> </li> <li id="cite_note-FOOTNOTETankovCont200395-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETankovCont200395_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTankovCont2003">Tankov & Cont 2003</a>, p. 95.</span> </li> <li id="cite_note-FOOTNOTETankovCont2003458-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETankovCont2003458_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTankovCont2003">Tankov & Cont 2003</a>, p. 458.</span> </li> <li id="cite_note-FOOTNOTESato199963-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESato199963_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSato1999">Sato 1999</a>, p. 63.</span> </li> <li id="cite_note-FOOTNOTECarrTorricelli2021698-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECarrTorricelli2021698_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCarrTorricelli2021">Carr & Torricelli 2021</a>, p. 698.</span> </li> <li id="cite_note-FOOTNOTEAzzoneBaviera2022503-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEAzzoneBaviera2022503_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEAzzoneBaviera2022503_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFAzzoneBaviera2022">Azzone & Baviera 2022</a>, p. 503.</span> </li> <li id="cite_note-FOOTNOTELiLiMendoza-Arriaga20165–6-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELiLiMendoza-Arriaga20165–6_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLiLiMendoza-Arriaga2016">Li, Li & Mendoza-Arriaga 2016</a>, pp. 5–6.</span> </li> <li id="cite_note-FOOTNOTELiLiMendoza-Arriaga20161-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELiLiMendoza-Arriaga20161_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLiLiMendoza-Arriaga2016">Li, Li & Mendoza-Arriaga 2016</a>, p. 1.</span> </li> <li id="cite_note-FOOTNOTEEberleinMadan20095-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEberleinMadan20095_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEberleinMadan2009">Eberlein & Madan 2009</a>, p. 5.</span> </li> <li id="cite_note-FOOTNOTECarrGemanMadanYor200739-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECarrGemanMadanYor200739_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCarrGemanMadanYor2007">Carr et al. 2007</a>, p. 39.</span> </li> <li id="cite_note-FOOTNOTEEberleinMadan2009-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEberleinMadan2009_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEberleinMadan2009">Eberlein & Madan 2009</a>.</span> </li> <li id="cite_note-FOOTNOTEEberleinMadan200919-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEberleinMadan200919_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEberleinMadan2009">Eberlein & Madan 2009</a>, p. 19.</span> </li> <li id="cite_note-FOOTNOTEBallottaKyriakou20141-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBallottaKyriakou20141_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBallottaKyriakou2014">Ballotta & Kyriakou 2014</a>, p. 1.</span> </li> <li id="cite_note-FOOTNOTEAzzoneBaviera20231–5-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAzzoneBaviera20231–5_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAzzoneBaviera2023">Azzone & Baviera 2023</a>, pp. 1–5.</span> </li> <li id="cite_note-FOOTNOTECarrGemanMadanYor200732-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECarrGemanMadanYor200732_21-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCarrGemanMadanYor2007">Carr et al. 2007</a>, p. 32.</span> </li> <li id="cite_note-FOOTNOTECarrGemanMadanYor200737-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECarrGemanMadanYor200737_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCarrGemanMadanYor2007">Carr et al. 2007</a>, p. 37.</span> </li> <li id="cite_note-FOOTNOTECarrGemanMadanYor200739–42-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTECarrGemanMadanYor200739–42_23-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFCarrGemanMadanYor2007">Carr et al. 2007</a>, pp. 39–42.</span> </li> <li id="cite_note-FOOTNOTEAzzoneBaviera2022506–508-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAzzoneBaviera2022506–508_24-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAzzoneBaviera2022">Azzone & Baviera 2022</a>, pp. 506–508.</span> </li> <li id="cite_note-FOOTNOTELiLiMendoza-Arriaga20163-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELiLiMendoza-Arriaga20163_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLiLiMendoza-Arriaga2016">Li, Li & Mendoza-Arriaga 2016</a>, pp. 3.</span> </li> <li id="cite_note-FOOTNOTELiLiMendoza-Arriaga201617-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELiLiMendoza-Arriaga201617_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLiLiMendoza-Arriaga2016">Li, Li & Mendoza-Arriaga 2016</a>, p. 17.</span> </li> <li id="cite_note-FOOTNOTELiLiZhang20171-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELiLiZhang20171_27-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLiLiZhang2017">Li, Li & Zhang 2017</a>, p. 1.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Additive_process&action=edit&section=19" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFTankovCont2003" class="citation book cs1">Tankov, Peter; Cont, Rama (2003). <i>Financial modelling with jump processes</i>. Chapman and Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1584884134" title="Special:BookSources/1584884134"><bdi>1584884134</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+modelling+with+jump+processes&rft.pub=Chapman+and+Hall&rft.date=2003&rft.isbn=1584884134&rft.aulast=Tankov&rft.aufirst=Peter&rft.au=Cont%2C+Rama&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAdditive+process" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSato1999" class="citation book cs1">Sato, Ken-Ito (1999). <i>Lévy processes and infinitely divisible distributions</i>. 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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 class="mw-selflink selflink">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 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Additive process - Wikipedia