Optional stopping theorem

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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">A martingale's expected value at a stopping time equals its initial expected value</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Optimal_stopping" title="Optimal stopping">Optimal stopping</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Optional_stopping_theorem" title="Special:EditPage/Optional stopping theorem">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Optional+stopping+theorem%22">"Optional stopping theorem"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Optional+stopping+theorem%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Optional+stopping+theorem%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Optional+stopping+theorem%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Optional+stopping+theorem%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Optional+stopping+theorem%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">February 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, the <b>optional stopping theorem</b> (or sometimes <b>Doob's optional sampling theorem</b>, for American probabilist <a href="/wiki/Joseph_L._Doob" title="Joseph L. Doob">Joseph Doob</a>) says that, under certain conditions, the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of a <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a> at a <a href="/wiki/Stopping_time" title="Stopping time">stopping time</a> is equal to its initial expected value. Since martingales can be used to model the wealth of a gambler participating in a fair game, the optional stopping theorem says that, on average, nothing can be gained by stopping play based on the information obtainable so far (i.e., without looking into the future). Certain conditions are necessary for this result to hold true. In particular, the theorem applies to <a href="/wiki/Martingale_(betting_system)" title="Martingale (betting system)">doubling strategies</a>. </p><p>The optional stopping theorem is an important tool of <a href="/wiki/Mathematical_finance" title="Mathematical finance">mathematical finance</a> in the context of the <a href="/wiki/Fundamental_theorem_of_asset_pricing" title="Fundamental theorem of asset pricing">fundamental theorem of asset pricing</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statement">Statement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optional_stopping_theorem&action=edit&section=1" title="Edit section: Statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A discrete-time version of the theorem is given below, with <span class="texhtml"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub></span> denoting the set of natural integers, including zero. </p><p>Let <span class="texhtml"><i>X</i> = (<i>X<sub>t</sub></i>)<sub><i>t</i>∈<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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub></sub></span> be a discrete-time <a href="/wiki/Martingale_(probability_theory)" title="Martingale (probability theory)">martingale</a> and <span class="texhtml"><i>τ</i></span> a <a href="/wiki/Stopping_time" title="Stopping time">stopping time</a> with values in <span class="texhtml"><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub> ∪ {∞</span>}, both with respect to a <a href="/wiki/Filtration_(probability_theory)" title="Filtration (probability theory)">filtration</a> <span class="texhtml">(<span class="mathcal" style="font-family: 'Lucida Calligraphy', 'Monotype Corsiva', 'URW Chancery L', 'Apple Chancery', 'Tex Gyre Chorus', cursive, serif;">F</span><sub><i>t</i></sub>)<sub><i>t</i>∈<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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub></sub></span>. Assume that one of the following three conditions holds: </p> <dl><dd>(<span id="math_a" class="reference nourlexpansion" style="font-weight:bold;">a</span>) The stopping time <span class="texhtml"><i>τ</i></span> is <a href="/wiki/Almost_surely" title="Almost surely">almost surely</a> bounded, i.e., there exists a <a href="/wiki/Mathematical_constant" title="Mathematical constant">constant</a> <span class="texhtml"><i>c</i> ∈ <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span></span> such that <span class="texhtml"><i>τ</i> ≤ <i>c</i></span> a.s.</dd> <dd>(<span id="math_b" class="reference nourlexpansion" style="font-weight:bold;">b</span>) The stopping time <span class="texhtml"><i>τ</i></span> has finite expectation and the conditional expectations of the absolute value of the martingale increments are almost surely bounded, more precisely, <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 {E} [\tau ]<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>τ</mi> <mo stretchy="false">]</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [\tau ]<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74d6be51b445ab7e5d8f9e7f6e686e284e66c267" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.468ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [\tau ]<\infty }"></span> and there exists a constant <span class="texhtml"><i>c</i></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 \mathbb {E} {\bigl [}|X_{t+1}-X_{t}|\,{\big \vert }\,{\mathcal {F}}_{t}{\bigr ]}\leq c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>≤</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} {\bigl [}|X_{t+1}-X_{t}|\,{\big \vert }\,{\mathcal {F}}_{t}{\bigr ]}\leq c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a011914ea6942c7a86e93b2b1675ce1c08f839" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.248ex; height:3.176ex;" alt="{\displaystyle \mathbb {E} {\bigl [}|X_{t+1}-X_{t}|\,{\big \vert }\,{\mathcal {F}}_{t}{\bigr ]}\leq c}"></span> almost surely on the event <span class="texhtml">{<i>τ</i> > <i>t</i></span>} for all <span class="texhtml"><i>t</i> ∈ <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub></span>.</dd> <dd>(<span id="math_c" class="reference nourlexpansion" style="font-weight:bold;">c</span>) There exists a constant <span class="texhtml"><i>c</i></span> such that <span class="texhtml">|<i>X</i><sub><i>t</i>∧<i>τ</i></sub>| ≤ <i>c</i></span> a.s. for all <span class="texhtml"><i>t</i> ∈ <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub></span> where <span class="texhtml">∧</span> denotes the <a href="/wiki/Join_and_meet" title="Join and meet">minimum operator</a>.</dd></dl> <p>Then <span class="texhtml"><i>X<sub>τ</sub></i></span> is an almost surely well defined random variable 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 \mathbb {E} [X_{\tau }]=\mathbb {E} [X_{0}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X_{\tau }]=\mathbb {E} [X_{0}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654e2434a6852a5adbf0d6e388134288ec6973cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.418ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [X_{\tau }]=\mathbb {E} [X_{0}].}"></span> </p><p>Similarly, if the stochastic process <span class="texhtml"><i>X</i> = (<i>X<sub>t</sub></i>)<sub><i>t</i>∈<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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub></sub></span> is a <a href="/wiki/Submartingale" class="mw-redirect" title="Submartingale">submartingale</a> or a <a href="/wiki/Supermartingale" class="mw-redirect" title="Supermartingale">supermartingale</a> and one of the above conditions holds, then </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 \mathbb {E} [X_{\tau }]\geq \mathbb {E} [X_{0}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X_{\tau }]\geq \mathbb {E} [X_{0}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb36174beedeac16b14313173fcd194f53815ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.418ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [X_{\tau }]\geq \mathbb {E} [X_{0}],}"></span></dd></dl> <p>for a submartingale, and </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 \mathbb {E} [X_{\tau }]\leq \mathbb {E} [X_{0}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X_{\tau }]\leq \mathbb {E} [X_{0}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5308f7c5229c2b376f1a572370ac8cf72043824f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.418ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [X_{\tau }]\leq \mathbb {E} [X_{0}],}"></span></dd></dl> <p>for a supermartingale. </p> <div class="mw-heading mw-heading3"><h3 id="Remark">Remark</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optional_stopping_theorem&action=edit&section=2" title="Edit section: Remark"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Under condition (<b><a href="#math_c">c</a></b>) it is possible that <span class="texhtml"><i>τ</i> = ∞</span> happens with positive probability. On this event <span class="texhtml"><i>X<sub>τ</sub></i></span> is defined as the almost surely existing pointwise limit of <span class="texhtml">(<i>X<sub>t</sub></i>)<sub><i>t</i>∈<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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub></sub></span> , see the proof below for details. </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=Optional_stopping_theorem&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Optional_stopping_theorem" title="Special:EditPage/Optional stopping theorem">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">May 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <ul><li>The optional stopping theorem can be used to prove the impossibility of successful betting strategies for a gambler with a finite lifetime (which gives condition (<b><a href="#math_a">a</a></b>)) or a house limit on bets (condition (<b><a href="#math_b">b</a></b>)). Suppose that the gambler can wager up to <i>c</i> dollars on a fair coin flip at times 1, 2, 3, etc., winning his wager if the coin comes up heads and losing it if the coin comes up tails. Suppose further that he can quit whenever he likes, but cannot predict the outcome of gambles that haven't happened yet. Then the gambler's fortune over time is a martingale, and the time <span class="texhtml"><i>τ</i></span> at which he decides to quit (or goes broke and is forced to quit) is a stopping time. So the theorem says that <span class="texhtml">E[<i>X<sub>τ</sub></i>] = E[<i>X</i><sub>0</sub>]</span>. In other words, the gambler leaves with the same amount of money <i>on average</i> as when he started. (The same result holds if the gambler, instead of having a house limit on individual bets, has a finite limit on his line of credit or how far in debt he may go, though this is easier to show with another version of the theorem.)</li> <li>Suppose a <a href="/wiki/Random_walk" title="Random walk">random walk</a> starting at <span class="texhtml"><i>a</i> ≥ 0</span> that goes up or down by one with equal probability on each step. Suppose further that the walk stops if it reaches <span class="texhtml">0</span> or <span class="texhtml"><i>m</i> ≥ <i>a</i></span>; the time at which this first occurs is a stopping time. If it is known that the expected time at which the walk ends is finite (say, from <a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a> theory), the optional stopping theorem predicts that the expected stop position is equal to the initial position <span class="texhtml"><i>a</i></span>. Solving <span class="texhtml"><i>a</i> = <i>pm</i> + (1 – <i>p</i>)0</span> for the probability <span class="texhtml"><i>p</i></span> that the walk reaches <span class="texhtml"><i>m</i></span> before <span class="texhtml">0</span> gives <span class="texhtml"><i>p</i> = <i>a</i>/<i>m</i></span>.</li> <li>Now consider a random walk <span class="texhtml"><i>X</i></span> that starts at <span class="texhtml">0</span> and stops if it reaches <span class="texhtml">–<i>m</i></span> or <span class="texhtml">+<i>m</i></span>, and use the <span class="texhtml"><i>Y<sub>n</sub></i> = <i>X</i><sub><i>n</i></sub><sup>2</sup> – <i>n</i></span> martingale from the <a href="/wiki/Martingale_(probability_theory)#Examples_of_martingales" title="Martingale (probability theory)">examples section</a>. If <span class="texhtml"><i>τ</i></span> is the time at which <span class="texhtml"><i>X</i></span> first reaches <span class="texhtml">±<i>m</i></span>, then <span class="texhtml">0 = E[<i>Y</i><sub>0</sub>] = E[<i>Y<sub>τ</sub></i>] = <i>m</i><sup>2</sup> – E[τ]</span>. This gives <span class="texhtml">E[<i>τ</i>] = <i>m</i><sup>2</sup></span>.</li> <li>Care must be taken, however, to ensure that one of the conditions of the theorem hold. For example, suppose the last example had instead used a 'one-sided' stopping time, so that stopping only occurred at <span class="texhtml">+<i>m</i></span>, not at <span class="texhtml">−<i>m</i></span>. The value of <span class="texhtml"><i>X</i></span> at this stopping time would therefore be <span class="texhtml"><i>m</i></span>. Therefore, the expectation value <span class="texhtml">E[<i>X<sub>τ</sub></i>]</span> must also be <span class="texhtml"><i>m</i></span>, seemingly in violation of the theorem which would give <span class="texhtml">E[<i>X<sub>τ</sub></i>] = 0</span>. The failure of the optional stopping theorem shows that all three of the conditions fail.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optional_stopping_theorem&action=edit&section=4" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml"><i>X<sup>τ</sup></i></span> denote the <a href="/wiki/Stopped_process" title="Stopped process">stopped process</a>, it is also a martingale (or a submartingale or supermartingale, respectively). Under condition (<b><a href="#math_a">a</a></b>) or (<b><a href="#math_b">b</a></b>), the random variable <span class="texhtml"><i>X<sub>τ</sub></i></span> is well defined. Under condition (<b><a href="#math_c">c</a></b>) the stopped process <span class="texhtml"><i>X<sup>τ</sup></i></span> is bounded, hence by Doob's <a href="/wiki/Martingale_convergence_theorem" class="mw-redirect" title="Martingale convergence theorem">martingale convergence theorem</a> it converges a.s. pointwise to a random variable which we call <span class="texhtml"><i>X<sub>τ</sub></i></span>. </p><p>If condition (<b><a href="#math_c">c</a></b>) holds, then the stopped process <span class="texhtml"><i>X<sup>τ</sup></i></span> is bounded by the constant random variable <span class="texhtml"><i>M</i> := <i>c</i></span>. Otherwise, writing the stopped process 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 X_{t}^{\tau }=X_{0}+\sum _{s=0}^{\tau -1\land t-1}(X_{s+1}-X_{s}),\quad t\in {\mathbb {N} }_{0},}"> <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"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mo>−</mo> <mn>1</mn> <mo>∧</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}^{\tau }=X_{0}+\sum _{s=0}^{\tau -1\land t-1}(X_{s+1}-X_{s}),\quad t\in {\mathbb {N} }_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129051d03f76295482f23dbb240cc73b00ca8e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.759ex; height:7.343ex;" alt="{\displaystyle X_{t}^{\tau }=X_{0}+\sum _{s=0}^{\tau -1\land t-1}(X_{s+1}-X_{s}),\quad t\in {\mathbb {N} }_{0},}"></span></dd></dl> <p>gives <span class="texhtml">|<i>X</i><sub><i>t</i></sub><sup><i>τ</i></sup>| ≤ <i>M</i></span> for all <span class="texhtml"><i>t</i> ∈ <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" 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 {N} }"></span><sub>0</sub></span>, 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 M:=|X_{0}|+\sum _{s=0}^{\tau -1}|X_{s+1}-X_{s}|=|X_{0}|+\sum _{s=0}^{\infty }|X_{s+1}-X_{s}|\cdot \mathbf {1} _{\{\tau >s\}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>></mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M:=|X_{0}|+\sum _{s=0}^{\tau -1}|X_{s+1}-X_{s}|=|X_{0}|+\sum _{s=0}^{\infty }|X_{s+1}-X_{s}|\cdot \mathbf {1} _{\{\tau >s\}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b856ca143c8df36d3414a9580a5138a7713b2cc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.966ex; height:7.343ex;" alt="{\displaystyle M:=|X_{0}|+\sum _{s=0}^{\tau -1}|X_{s+1}-X_{s}|=|X_{0}|+\sum _{s=0}^{\infty }|X_{s+1}-X_{s}|\cdot \mathbf {1} _{\{\tau >s\}}}"></span>.</dd></dl> <p>By the <a href="/wiki/Monotone_convergence_theorem" title="Monotone convergence theorem">monotone convergence theorem</a> </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 \mathbb {E} [M]=\mathbb {E} [|X_{0}|]+\sum _{s=0}^{\infty }\mathbb {E} {\bigl [}|X_{s+1}-X_{s}|\cdot \mathbf {1} _{\{\tau >s\}}{\bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>M</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>></mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [M]=\mathbb {E} [|X_{0}|]+\sum _{s=0}^{\infty }\mathbb {E} {\bigl [}|X_{s+1}-X_{s}|\cdot \mathbf {1} _{\{\tau >s\}}{\bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949b2c97ece2dfc78f5b7c95ee886575cf4d3880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.454ex; height:6.843ex;" alt="{\displaystyle \mathbb {E} [M]=\mathbb {E} [|X_{0}|]+\sum _{s=0}^{\infty }\mathbb {E} {\bigl [}|X_{s+1}-X_{s}|\cdot \mathbf {1} _{\{\tau >s\}}{\bigr ]}}"></span>.</dd></dl> <p>If condition (<b><a href="#math_a">a</a></b>) holds, then this series only has a finite number of non-zero terms, hence <span class="texhtml"><i>M</i></span> is integrable. </p><p>If condition (<b><a href="#math_b">b</a></b>) holds, then we continue by inserting a <a href="/wiki/Conditional_expectation" title="Conditional expectation">conditional expectation</a> and using that the event <span class="texhtml">{<i>τ</i> > <i>s</i></span>} is known at time <span class="texhtml"><i>s</i></span> (note that <span class="texhtml"><i>τ</i></span> is assumed to be a stopping time with respect to the filtration), hence </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 {\begin{aligned}\mathbb {E} [M]&=\mathbb {E} [|X_{0}|]+\sum _{s=0}^{\infty }\mathbb {E} {\bigl [}\underbrace {\mathbb {E} {\bigl [}|X_{s+1}-X_{s}|{\big |}{\mathcal {F}}_{s}{\bigr ]}\cdot \mathbf {1} _{\{\tau >s\}}} _{\leq \,c\,\mathbf {1} _{\{\tau >s\}}{\text{ a.s. by (b)}}}{\bigr ]}\\&\leq \mathbb {E} [|X_{0}|]+c\sum _{s=0}^{\infty }\mathbb {P} (\tau >s)\\&=\mathbb {E} [|X_{0}|]+c\,\mathbb {E} [\tau ]<\infty ,\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>M</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>></mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≤</mo> <mspace width="thinmathspace" /> <mi>c</mi> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>τ</mi> <mo>></mo> <mi>s</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> a.s. by (b)</mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mi>c</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>></mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">]</mo> <mo>+</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <mi>τ</mi> <mo stretchy="false">]</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbb {E} [M]&=\mathbb {E} [|X_{0}|]+\sum _{s=0}^{\infty }\mathbb {E} {\bigl [}\underbrace {\mathbb {E} {\bigl [}|X_{s+1}-X_{s}|{\big |}{\mathcal {F}}_{s}{\bigr ]}\cdot \mathbf {1} _{\{\tau >s\}}} _{\leq \,c\,\mathbf {1} _{\{\tau >s\}}{\text{ a.s. by (b)}}}{\bigr ]}\\&\leq \mathbb {E} [|X_{0}|]+c\sum _{s=0}^{\infty }\mathbb {P} (\tau >s)\\&=\mathbb {E} [|X_{0}|]+c\,\mathbb {E} [\tau ]<\infty ,\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a3abdf4111f2be51565da63ff53ccac4af5b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:53.017ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}\mathbb {E} [M]&=\mathbb {E} [|X_{0}|]+\sum _{s=0}^{\infty }\mathbb {E} {\bigl [}\underbrace {\mathbb {E} {\bigl [}|X_{s+1}-X_{s}|{\big |}{\mathcal {F}}_{s}{\bigr ]}\cdot \mathbf {1} _{\{\tau >s\}}} _{\leq \,c\,\mathbf {1} _{\{\tau >s\}}{\text{ a.s. by (b)}}}{\bigr ]}\\&\leq \mathbb {E} [|X_{0}|]+c\sum _{s=0}^{\infty }\mathbb {P} (\tau >s)\\&=\mathbb {E} [|X_{0}|]+c\,\mathbb {E} [\tau ]<\infty ,\\\end{aligned}}}"></span></dd></dl> <p>where a <a href="/wiki/Expected_value#Discrete_distribution_taking_only_non-negative_integer_values" title="Expected value">representation of the expected value of non-negative integer-valued random variables</a> is used for the last equality. </p><p>Therefore, under any one of the three conditions in the theorem, the stopped process is dominated by an integrable random variable <span class="texhtml"><i>M</i></span>. Since the stopped process <span class="texhtml"><i>X<sup>τ</sup></i></span> converges almost surely to <span class="texhtml"><i>X<sub>τ</sub></i></span>, the <a href="/wiki/Dominated_convergence_theorem" title="Dominated convergence theorem">dominated convergence theorem</a> implies </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 \mathbb {E} [X_{\tau }]=\lim _{t\to \infty }\mathbb {E} [X_{t}^{\tau }].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X_{\tau }]=\lim _{t\to \infty }\mathbb {E} [X_{t}^{\tau }].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5e031217ea6546505d9fab12e7bfbbb14b08c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.786ex; height:4.009ex;" alt="{\displaystyle \mathbb {E} [X_{\tau }]=\lim _{t\to \infty }\mathbb {E} [X_{t}^{\tau }].}"></span></dd></dl> <p>By the martingale property of the stopped process, </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 \mathbb {E} [X_{t}^{\tau }]=\mathbb {E} [X_{0}],\quad t\in {\mathbb {N} }_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X_{t}^{\tau }]=\mathbb {E} [X_{0}],\quad t\in {\mathbb {N} }_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97429b7052d07a60cb537f6365b1807d73c2f101" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.26ex; height:3.009ex;" alt="{\displaystyle \mathbb {E} [X_{t}^{\tau }]=\mathbb {E} [X_{0}],\quad t\in {\mathbb {N} }_{0},}"></span></dd></dl> <p>hence </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 \mathbb {E} [X_{\tau }]=\mathbb {E} [X_{0}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [X_{\tau }]=\mathbb {E} [X_{0}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654e2434a6852a5adbf0d6e388134288ec6973cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.418ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [X_{\tau }]=\mathbb {E} [X_{0}].}"></span></dd></dl> <p>Similarly, if <span class="texhtml"><i>X</i></span> is a submartingale or supermartingale, respectively, change the equality in the last two formulas to the appropriate inequality. </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=Optional_stopping_theorem&action=edit&section=5" 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> <ol><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="CITEREFGrimmettStirzaker2001" class="citation book cs1">Grimmett, Geoffrey R.; Stirzaker, David R. (2001). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/probabilityrando00grgr"><i>Probability and Random Processes</i></a></span> (3rd ed.). Oxford University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/probabilityrando00grgr/page/n501">491</a>–495. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780198572220" title="Special:BookSources/9780198572220"><bdi>9780198572220</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Random+Processes&rft.pages=491-495&rft.edition=3rd&rft.pub=Oxford+University+Press&rft.date=2001&rft.isbn=9780198572220&rft.aulast=Grimmett&rft.aufirst=Geoffrey+R.&rft.au=Stirzaker%2C+David+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprobabilityrando00grgr&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptional+stopping+theorem" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhattacharyaWaymire2007" class="citation book cs1">Bhattacharya, Rabi; Waymire, Edward C. (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=clQQVdUNBmcC&q=%22Optional+stopping%22"><i>A Basic Course in Probability Theory</i></a>. Springer. pp. <span class="nowrap">43–</span>45. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-71939-9" title="Special:BookSources/978-0-387-71939-9"><bdi>978-0-387-71939-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Basic+Course+in+Probability+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E43-%3C%2Fspan%3E45&rft.pub=Springer&rft.date=2007&rft.isbn=978-0-387-71939-9&rft.aulast=Bhattacharya&rft.aufirst=Rabi&rft.au=Waymire%2C+Edward+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DclQQVdUNBmcC%26q%3D%2522Optional%2Bstopping%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOptional+stopping+theorem" class="Z3988"></span></li></ol> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Optional_stopping_theorem&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://math.mit.edu/~sheffield/martingalenote.pdf">Doob's Optional Stopping Theorem</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" 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=Optional_stopping_theorem&oldid=1273633382">https://en.wikipedia.org/w/index.php?title=Optional_stopping_theorem&oldid=1273633382</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:Probability_theorems" title="Category:Probability theorems">Probability theorems</a></li><li><a href="/wiki/Category:Theorems_in_statistics" title="Category:Theorems in statistics">Theorems in statistics</a></li><li><a href="/wiki/Category:Martingale_theory" title="Category:Martingale theory">Martingale theory</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:Articles_needing_additional_references_from_February_2012" title="Category:Articles needing additional references from February 2012">Articles needing additional references from February 2012</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_May_2024" title="Category:Articles needing additional references from May 2024">Articles needing additional references from May 2024</a></li><li><a href="/wiki/Category:Articles_containing_proofs" title="Category:Articles containing proofs">Articles containing proofs</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 3 February 2025, at 08:07<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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Optional stopping theorem - Wikipedia