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class="vector-toc-list"> <li id="toc-Discrete_time" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_time"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Discrete time</span> </div> </a> <ul id="toc-Discrete_time-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_case" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_case"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>General case</span> </div> </a> <ul id="toc-General_case-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_adapted_process" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_adapted_process"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>As adapted process</span> </div> </a> <ul id="toc-As_adapted_process-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comments"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Comments</span> </div> </a> <ul id="toc-Comments-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Localization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Localization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Localization</span> </div> </a> <ul id="toc-Localization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types_of_stopping_times" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Types_of_stopping_times"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Types of stopping times</span> </div> </a> <ul id="toc-Types_of_stopping_times-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stopping_rules_in_clinical_trials" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stopping_rules_in_clinical_trials"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Stopping rules in clinical trials</span> </div> </a> <ul id="toc-Stopping_rules_in_clinical_trials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul 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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the musical phenomenon, see <a href="/wiki/Stop-time" title="Stop-time">stop-time</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:HittingTimes1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/HittingTimes1.png/300px-HittingTimes1.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/HittingTimes1.png/450px-HittingTimes1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/HittingTimes1.png/600px-HittingTimes1.png 2x" data-file-width="1200" data-file-height="900" /></a><figcaption>Example of a stopping time: a <a href="/wiki/Hitting_time" title="Hitting time">hitting time</a> of <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>. The process starts at 0 and is stopped as soon as it hits 1.</figcaption></figure> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, in particular in the study of <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a>, a <b>stopping time</b> (also <b>Markov time</b>, <b>Markov moment</b>, <b>optional stopping time</b> or <b>optional time</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>) is a specific type of “random time”: a <a href="/wiki/Random_variable" title="Random variable">random variable</a> whose value is interpreted as the time at which a given stochastic process exhibits a certain behavior of interest. A stopping time is often defined by a <b>stopping rule</b>, a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will <a href="/wiki/Almost_always" class="mw-redirect" title="Almost always">almost always</a> lead to a decision to stop at some finite time. </p><p>Stopping times occur in <a href="/wiki/Decision_theory" title="Decision theory">decision theory</a>, and the <a href="/wiki/Optional_stopping_theorem" title="Optional stopping theorem">optional stopping theorem</a> is an important result in this context. Stopping times are also frequently applied in mathematical proofs to “tame the continuum of time”, as Chung put it in his book (1982). </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=Stopping_time&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Discrete_time">Discrete time</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=2" title="Edit section: Discrete time"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> be a random variable, which is defined on the <a href="/wiki/Filtered_probability_space" class="mw-redirect" title="Filtered probability space">filtered probability space</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 (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac313fbd9cfd5037f077a803c4544773dd27de46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.461ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n\in \mathbb {N} },P)}"></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 {N} \cup \{+\infty \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \cup \{+\infty \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba4cb4b2101ef1411bf0fd221cad064883cdfa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.717ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \cup \{+\infty \}}"></span>. Then <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is called a stopping time (with respect to the <a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">filtration</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 \mathbb {F} =(({\mathcal {F}}_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} =(({\mathcal {F}}_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76f9f4a35fc0a8ad59979156e8617acf4314980d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.624ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} =(({\mathcal {F}}_{n})_{n\in \mathbb {N} }}"></span>), if the following condition holds: </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 \{\tau =n\}\in {\mathcal {F}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>=</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\tau =n\}\in {\mathcal {F}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14edfab5b4db08e7abc1e7a33d94babaed1352a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.75ex; height:2.843ex;" alt="{\displaystyle \{\tau =n\}\in {\mathcal {F}}_{n}}"></span> 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></dd></dl> <p>Intuitively, this condition means that the "decision" of whether to stop 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> must be based only on the information present 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, not on any future information. </p> <div class="mw-heading mw-heading3"><h3 id="General_case">General case</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=3" title="Edit section: General case"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> be a random variable, which is defined on the <a href="/wiki/Filtered_probability_space" class="mw-redirect" title="Filtered probability space">filtered probability space</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 (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in T},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in T},P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231e94fd5fa96bcaacbe06e49bf0d5648292123f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.647ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in T},P)}"></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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. In most cases, <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,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=[0,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e46208559fcdc4616f0172fe08e70282fa1d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.614ex; height:2.843ex;" alt="{\displaystyle T=[0,+\infty )}"></span>. Then <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is called a stopping time (with respect to the <a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">filtration</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 \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd36fc1fa46db611af5890d9caa11dd6875ff53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.904ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}"></span>), if the following condition holds: </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 \{\tau \leq t\}\in {\mathcal {F}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>≤</mo> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\tau \leq t\}\in {\mathcal {F}}_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6c99390c54db1e2222c4f00e3efcad59695245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.803ex; height:2.843ex;" alt="{\displaystyle \{\tau \leq t\}\in {\mathcal {F}}_{t}}"></span> 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\in T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4fe93f70df3818ecca67c2ca44f087483951856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.317ex; height:2.176ex;" alt="{\displaystyle t\in T}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="As_adapted_process">As adapted process</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=4" title="Edit section: As adapted process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> be a random variable, which is defined on the <a href="/wiki/Filtered_probability_space" class="mw-redirect" title="Filtered probability space">filtered probability space</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 (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in T},P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in T},P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231e94fd5fa96bcaacbe06e49bf0d5648292123f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.647ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in T},P)}"></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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. Then <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is called a stopping time if the <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=(X_{t})_{t\in T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=(X_{t})_{t\in T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/698b49d3abb3b2dfff77d077a5da132725ec1ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.717ex; height:2.843ex;" alt="{\displaystyle X=(X_{t})_{t\in T}}"></span>, defined by </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 X_{t}:={\begin{cases}1&{\text{ if }}t<\tau \\0&{\text{ if }}t\geq \tau \end{cases}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>t</mi> <mo><</mo> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>t</mi> <mo>≥</mo> <mi>τ</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}:={\begin{cases}1&{\text{ if }}t<\tau \\0&{\text{ if }}t\geq \tau \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aff34473286897d248c3c8f2deb9127d016f4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.135ex; height:6.176ex;" alt="{\displaystyle X_{t}:={\begin{cases}1&{\text{ if }}t<\tau \\0&{\text{ if }}t\geq \tau \end{cases}}}"></span></dd></dl> <p>is <a href="/wiki/Adapted_process" title="Adapted process">adapted</a> to the filtration <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 {F} =({\mathcal {F}}_{t})_{t\in T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>∈</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd36fc1fa46db611af5890d9caa11dd6875ff53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.904ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\in T}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Comments">Comments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=5" title="Edit section: Comments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some authors explicitly exclude cases 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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> can be <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 +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"></span>, whereas other authors allow <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> to take any value in the closure 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}"> <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=6" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To illustrate some examples of random times that are stopping rules and some that are not, consider a gambler playing <a href="/wiki/Roulette" title="Roulette">roulette</a> with a typical house edge, starting with $100 and betting $1 on red in each game: </p> <ul><li>Playing exactly five games corresponds to the stopping time <i>τ</i> = 5, and <i>is</i> a stopping rule.</li> <li>Playing until they either run out of money or have played 500 games <i>is</i> a stopping rule.</li> <li>Playing until they obtain the maximum amount ahead they will ever be <i>is not</i> a stopping rule and does not provide a stopping time, as it requires information about the future as well as the present and past.</li> <li>Playing until they double their money (borrowing if necessary) <i>is not</i> a stopping rule, as there is a positive probability that they will never double their money.<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="The text near this tag may need clarification or removal of jargon. (December 2022)">clarification needed</span></a> (see <a href="/wiki/Talk:Stopping_time#Examples" title="Talk:Stopping time">talk</a>)</i>]</sup></li> <li>Playing until they either double their money or run out of money <i>is</i> a stopping rule, even though there is potentially no limit to the number of games they play, since the probability that they stop in a finite time is 1.</li></ul> <p>To illustrate the more general definition of stopping time, consider <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a>, which is 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 (B_{t})_{t\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo 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 (B_{t})_{t\geq 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c39832102d3a9bfe8ab176ec6ad2243067ab7085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.326ex; height:2.843ex;" alt="{\displaystyle (B_{t})_{t\geq 0}}"></span>, where each <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_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f92553a3c8585519e540724dffe9306c83ca2f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.59ex; height:2.509ex;" alt="{\displaystyle B_{t}}"></span> is a random variable defined on the probability space <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 (\Omega ,{\mathcal {F}},\mathbb {P} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb8743f7565082ed1a9ee0490d9d71be82eafaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.902ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}"></span>. We define a filtration on this probability space by letting <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 {\mathcal {F}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.497ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{t}}"></span> be the <i>σ</i>-algebra generated by all the sets of the form <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_{s})^{-1}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (B_{s})^{-1}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d738e11c82bce7ccd26976e8a2e87b7157c95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.462ex; height:3.176ex;" alt="{\displaystyle (B_{s})^{-1}(A)}"></span> 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 0\leq s\leq t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>s</mi> <mo>≤</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq s\leq t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efd3155df471ed05348da5307e4104e085f77d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.289ex; height:2.343ex;" alt="{\displaystyle 0\leq s\leq 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 A\subseteq \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e286ff36e0ad1e0e1e70be50b1cbcfe232b94867" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.52ex; height:2.343ex;" alt="{\displaystyle A\subseteq \mathbb {R} }"></span> is a <a href="/wiki/Borel_set" title="Borel set">Borel set</a>. Intuitively, an event <i>E</i> is 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 {\mathcal {F}}_{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5209aeab53016ae720999b2a50e7f0bc27cb33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.497ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}_{t}}"></span> if and only if we can determine whether <i>E</i> is true or false just by observing the Brownian motion from time 0 to time <i>t</i>. </p> <ul><li>Every constant <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 \tau :=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>:=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau :=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea3af160378e56a5e5a849ca034bef5118eb78d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.841ex; height:2.343ex;" alt="{\displaystyle \tau :=t_{0}}"></span> is (trivially) a stopping time; it corresponds to the stopping rule "stop 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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </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/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}"></span>".</li> <li>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 a\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d8ec83fa9b414f3a72cb56cc841b8b76e22bec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.395ex; height:2.176ex;" alt="{\displaystyle a\in \mathbb {R} .}"></span> Then <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 \tau :=\inf\{t\geq 0\mid B_{t}=a\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>:=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo>∣</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau :=\inf\{t\geq 0\mid B_{t}=a\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242a5a15ab891810e2b6f409081feadc4ae46dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.879ex; height:2.843ex;" alt="{\displaystyle \tau :=\inf\{t\geq 0\mid B_{t}=a\}}"></span> is a stopping time for Brownian motion, corresponding to the stopping rule: "stop as soon as the Brownian motion hits the value <i>a</i>."</li> <li>Another stopping time is given by <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 \tau :=\inf\{t\geq 1\mid B_{s}>0{\text{ for all }}s\in [t-1,t]\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>:=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>t</mi> <mo>≥</mo> <mn>1</mn> <mo>∣</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau :=\inf\{t\geq 1\mid B_{s}>0{\text{ for all }}s\in [t-1,t]\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cef20dca0c1c0be36e4ef25ed376f986aa3349" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.914ex; height:2.843ex;" alt="{\displaystyle \tau :=\inf\{t\geq 1\mid B_{s}>0{\text{ for all }}s\in [t-1,t]\}}"></span>. It corresponds to the stopping rule "stop as soon as the Brownian motion has been positive over a contiguous stretch of length 1 time unit."</li> <li>In general, if τ<sub>1</sub> and τ<sub>2</sub> are stopping times 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 \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <msub> <mrow> <mo>{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd52ce527bf2d1efcbb5f07c02a9cc7adfe5977d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.005ex; height:3.343ex;" alt="{\displaystyle \left(\Omega ,{\mathcal {F}},\left\{{\mathcal {F}}_{t}\right\}_{t\geq 0},\mathbb {P} \right)}"></span> then their minimum <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 \tau _{1}\wedge \tau _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{1}\wedge \tau _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f349302caecfdd0348fe8d78ed6ea19b489d272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.723ex; height:2.343ex;" alt="{\displaystyle \tau _{1}\wedge \tau _{2}}"></span>, their maximum <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 \tau _{1}\vee \tau _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∨</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{1}\vee \tau _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7e82480332f9f4d65b6c72fed5beae70841d00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.723ex; height:2.343ex;" alt="{\displaystyle \tau _{1}\vee \tau _{2}}"></span>, and their sum <i>τ</i><sub>1</sub> + <i>τ</i><sub>2</sub> are also stopping times. (This is not true for differences and products, because these may require "looking into the future" to determine when to stop.)</li></ul> <p><a href="/wiki/Hitting_time" title="Hitting time">Hitting times</a> like the second example above can be important examples of stopping times. While it is relatively straightforward to show that essentially all stopping times are hitting times,<sup id="cite_ref-Fischer_(2013)_2-0" class="reference"><a href="#cite_note-Fischer_(2013)-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> it can be much more difficult to show that a certain hitting time is a stopping time. The latter types of results are known as the <a href="/wiki/Hitting_time#Début_theorem" title="Hitting time">Début theorem</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Localization">Localization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=7" title="Edit section: Localization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stopping times are frequently used to generalize certain properties of stochastic processes to situations in which the required property is satisfied in only a local sense. First, if <i>X</i> is a process and τ is a stopping time, then <i>X</i><sup><i>τ</i></sup> is used to denote the process <i>X</i> stopped at time <i>τ</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 X_{t}^{\tau }=X_{\min(t,\tau )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}^{\tau }=X_{\min(t,\tau )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/385375522ca82265cbe94a34978fb316971a0261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.255ex; height:3.009ex;" alt="{\displaystyle X_{t}^{\tau }=X_{\min(t,\tau )}}"></span></dd></dl> <p>Then, <i>X</i> is said to locally satisfy some property <i>P</i> if there exists a sequence of stopping times τ<sub><i>n</i></sub>, which increases to infinity and for which the processes </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 \mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d87030efad11eeb59410ed6230d83c9554541e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.909ex; height:3.176ex;" alt="{\displaystyle \mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}}"></span></dd></dl> <p>satisfy property <i>P</i>. Common examples, with time index set <i>I</i> = [0, ∞), are as follows: </p> <blockquote><p><b>Local martingale process</b>. A process <i>X</i> is a <a href="/wiki/Local_martingale" title="Local martingale">local martingale</a> if it is <a href="/wiki/C%C3%A0dl%C3%A0g" title="Càdlàg">càdlàg</a><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 appears to contradict the definition given in the article Local Martingale, where cadlag is not required. (October 2023)">clarification needed</span></a></i>]</sup> and there exists a sequence of stopping times τ<sub><i>n</i></sub> increasing to infinity, such that </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 \mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d87030efad11eeb59410ed6230d83c9554541e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.909ex; height:3.176ex;" alt="{\displaystyle \mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}}"></span></dd></dl><p> is a <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a> for each <i>n</i>.</p></blockquote> <blockquote><p><b>Locally integrable process</b>. A non-negative and increasing process <i>X</i> is locally integrable if there exists a sequence of stopping times <i>τ</i><sub><i>n</i></sub> increasing to infinity, such that </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[\mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}\right]<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">E</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[\mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}\right]<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc266b3f1da8d8851508175d3fb522e6ff56d73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.854ex; height:3.343ex;" alt="{\displaystyle \operatorname {E} \left[\mathbf {1} _{\{\tau _{n}>0\}}X^{\tau _{n}}\right]<\infty }"></span></dd></dl><p> for each <i>n</i>.</p></blockquote> <div class="mw-heading mw-heading2"><h2 id="Types_of_stopping_times">Types of stopping times</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=8" title="Edit section: Types of stopping times"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stopping times, with time index set <i>I</i> = [0,∞), are often divided into one of several types depending on whether it is possible to predict when they are about to occur. </p><p>A stopping time <i>τ</i> is <b>predictable</b> if it is equal to the limit of an increasing sequence of stopping times <i>τ</i><sub><i>n</i></sub> satisfying <i>τ</i><sub><i>n</i></sub> < <i>τ</i> whenever <i>τ</i> > 0. The sequence <i>τ</i><sub><i>n</i></sub> is said to <i>announce</i> <i>τ</i>, and predictable stopping times are sometimes known as <i>announceable</i>. Examples of predictable stopping times are <a href="/wiki/Hitting_time" title="Hitting time">hitting times</a> of continuous and <a href="/wiki/Adapted_process" title="Adapted process">adapted</a> processes. If <i>τ</i> is the first time at which a continuous and real valued process <i>X</i> is equal to some value <i>a</i>, then it is announced by the sequence <i>τ</i><sub><i>n</i></sub>, where <i>τ</i><sub><i>n</i></sub> is the first time at which <i>X</i> is within a distance of 1/<i>n</i> of <i>a</i>. </p><p><b>Accessible</b> stopping times are those that can be covered by a sequence of predictable times. That is, stopping time <i>τ</i> is accessible if, P(<i>τ</i> = <i>τ</i><sub><i>n</i></sub> for some <i>n</i>) = 1, where <i>τ</i><sub><i>n</i></sub> are predictable times. </p><p>A stopping time <i>τ</i> is <b>totally inaccessible</b> if it can never be announced by an increasing sequence of stopping times. Equivalently, P(<i>τ</i> = <i>σ</i> < ∞) = 0 for every predictable time <i>σ</i>. Examples of totally inaccessible stopping times include the jump times of <a href="/wiki/Poisson_process" class="mw-redirect" title="Poisson process">Poisson processes</a>. </p><p>Every stopping time <i>τ</i> can be uniquely decomposed into an accessible and totally inaccessible time. That is, there exists a unique accessible stopping time σ and totally inaccessible time υ such that <i>τ</i> = <i>σ</i> whenever <i>σ</i> < ∞, <i>τ</i> = <i>υ</i> whenever <i>υ</i> < ∞, and <i>τ</i> = ∞ whenever <i>σ</i> = <i>υ</i> = ∞. Note that in the statement of this decomposition result, stopping times do not have to be almost surely finite, and can equal ∞. </p> <div class="mw-heading mw-heading2"><h2 id="Stopping_rules_in_clinical_trials">Stopping rules in clinical trials</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=9" title="Edit section: Stopping rules in clinical trials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Clinical trials in medicine often perform interim analysis, in order to determine whether the trial has already met its endpoints. However, interim analysis create the risk of false-positive results, and therefore stopping boundaries are used to determine the number and timing of interim analysis (also known as alpha-spending, to denote the rate of false positives). At each of R interim tests, the trial is stopped if the likelihood is below a threshold p, which depends on the method used. See <a href="/wiki/Sequential_analysis" title="Sequential analysis">Sequential analysis</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Optimal_stopping" title="Optimal stopping">Optimal stopping</a></li> <li><a href="/wiki/Odds_algorithm" title="Odds algorithm">Odds algorithm</a></li> <li><a href="/wiki/Secretary_problem" title="Secretary problem">Secretary problem</a></li> <li><a href="/wiki/Hitting_time" title="Hitting time">Hitting time</a></li> <li><a href="/wiki/Stopped_process" title="Stopped process">Stopped process</a></li> <li><a href="/wiki/Disorder_problem" title="Disorder problem">Disorder problem</a></li> <li><a href="/wiki/Hitting_time#Début_theorem" title="Hitting time">Début theorem</a></li> <li><a href="/wiki/Sequential_analysis" title="Sequential analysis">Sequential analysis</a></li></ul> <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=Stopping_time&action=edit&section=11" 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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFKallenberg2017" class="citation book cs1"><a href="/wiki/Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, Olav</a> (2017). <i>Random Measures, Theory and Applications</i>. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. p. 347. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-41598-7">10.1007/978-3-319-41598-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-41596-3" title="Special:BookSources/978-3-319-41596-3"><bdi>978-3-319-41596-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Measures%2C+Theory+and+Applications&rft.place=Switzerland&rft.series=Probability+Theory+and+Stochastic+Modelling&rft.pages=347&rft.pub=Springer&rft.date=2017&rft_id=info%3Adoi%2F10.1007%2F978-3-319-41598-7&rft.isbn=978-3-319-41596-3&rft.aulast=Kallenberg&rft.aufirst=Olav&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStopping+time" class="Z3988"></span> </span> </li> <li id="cite_note-Fischer_(2013)-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Fischer_(2013)_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFischer2013" class="citation journal cs1">Fischer, Tom (2013). "On simple representations of stopping times and stopping time sigma-algebras". <i>Statistics and Probability Letters</i>. <b>83</b> (1): 345–349. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1112.1603">1112.1603</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.spl.2012.09.024">10.1016/j.spl.2012.09.024</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+and+Probability+Letters&rft.atitle=On+simple+representations+of+stopping+times+and+stopping+time+sigma-algebras&rft.volume=83&rft.issue=1&rft.pages=345-349&rft.date=2013&rft_id=info%3Aarxiv%2F1112.1603&rft_id=info%3Adoi%2F10.1016%2Fj.spl.2012.09.024&rft.aulast=Fischer&rft.aufirst=Tom&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStopping+time" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stopping_time&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Thomas_S._Ferguson" title="Thomas S. Ferguson">Thomas S. Ferguson</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080913145914/http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ss%2F1177012493">“Who solved the secretary problem?”, Stat. Sci. vol. 4, 282–296, (1989).</a></li> <li><a rel="nofollow" class="external text" href="http://www.bramdejonge.nl/pdf/stoppingtimes.pdf">An introduction to stopping times.</a></li> <li><a href="/wiki/F._Thomas_Bruss" class="mw-redirect" title="F. Thomas Bruss">F. Thomas Bruss</a>, “Sum the odds to one and stop”, Annals of Probability, Vol. 4, 1384–1391,(2000)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChung1982" class="citation book cs1">Chung, Kai Lai (1982). <i>Lectures from Markov processes to Brownian motion</i>. Grundlehren der Mathematischen Wissenschaften No. 249. New York, NY: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90618-8" title="Special:BookSources/978-0-387-90618-8"><bdi>978-0-387-90618-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+from+Markov+processes+to+Brownian+motion&rft.place=New+York%2C+NY&rft.series=Grundlehren+der+Mathematischen+Wissenschaften+No.+249&rft.pub=Springer-Verlag&rft.date=1982&rft.isbn=978-0-387-90618-8&rft.aulast=Chung&rft.aufirst=Kai+Lai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStopping+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFH._Vincent_Poor_and_Olympia_Hadjiliadis2008" class="citation book cs1">H. Vincent Poor and Olympia Hadjiliadis (2008). <i>Quickest Detection</i> (First ed.). Cambridge, England: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-62104-5" title="Special:BookSources/978-0-521-62104-5"><bdi>978-0-521-62104-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quickest+Detection&rft.place=Cambridge%2C+England&rft.edition=First&rft.pub=Cambridge+University+Press&rft.date=2008&rft.isbn=978-0-521-62104-5&rft.au=H.+Vincent+Poor+and+Olympia+Hadjiliadis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStopping+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProtter2005" class="citation book cs1">Protter, Philip E. (2005). <i>Stochastic integration and differential equations</i>. Stochastic Modelling and Applied Probability No. 21 (2nd edition (version 2.1, corrected 3rd printing) ed.). Berlin: Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-00313-7" title="Special:BookSources/978-3-540-00313-7"><bdi>978-3-540-00313-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stochastic+integration+and+differential+equations&rft.place=Berlin&rft.series=Stochastic+Modelling+and+Applied+Probability+No.+21&rft.edition=2nd+edition+%28version+2.1%2C+corrected+3rd+printing%29&rft.pub=Springer-Verlag&rft.date=2005&rft.isbn=978-3-540-00313-7&rft.aulast=Protter&rft.aufirst=Philip+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStopping+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShiryaev2007" class="citation book cs1"><a href="/wiki/Albert_Shiryaev" title="Albert Shiryaev">Shiryaev, Albert N.</a> (2007). <i>Optimal Stopping Rules</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-74010-0" title="Special:BookSources/978-3-540-74010-0"><bdi>978-3-540-74010-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimal+Stopping+Rules&rft.pub=Springer&rft.date=2007&rft.isbn=978-3-540-74010-0&rft.aulast=Shiryaev&rft.aufirst=Albert+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStopping+time" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Stopping_time&oldid=1257088207">https://en.wikipedia.org/w/index.php?title=Stopping_time&oldid=1257088207</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Stochastic_processes" title="Category:Stochastic processes">Stochastic processes</a></li><li><a href="/wiki/Category:Optimal_decisions" title="Category:Optimal decisions">Optimal decisions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_December_2022" title="Category:Wikipedia articles needing clarification from December 2022">Wikipedia articles needing clarification from December 2022</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_October_2023" title="Category:Wikipedia articles needing clarification from October 2023">Wikipedia articles needing clarification from October 2023</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 November 2024, at 04:36<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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