<!DOCTYPE html>
<html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-0 vector-toc-not-available vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-0 skin-theme-clientpref-day vector-sticky-header-enabled" lang="de" dir="ltr"><head>
<meta charset="UTF-8">
<title>Feynman-Kac-Formel</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="icon" type="image/png" href="./_res_/favicon.png">
<link rel="canonical" href="https://de.wikipedia.org/wiki/Feynman-Kac-Formel"> <link href="./_mw_/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.wikimediamessages.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./_mw_/skins.vector.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link href="./_mw_/ext.gadget.citeRef.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.defaultPlainlinks.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonHide.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonLayout.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiCommonStyle.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiDarkmode.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.dewikiResponsive.css" rel="stylesheet" type="text/css">
<link href="./_mw_/ext.gadget.specialSearch.css" rel="stylesheet" type="text/css">
<link rel="stylesheet" type="text/css" href="./_mw_/site.styles.css">
<link rel="stylesheet" type="text/css" href="./_mw_/noscript.css">
<link rel="stylesheet" type="text/css" href="./_res_/footer.css">
<link rel="stylesheet" type="text/css" href="./_res_/vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Feynman-Kac-Formel rootpage-Feynman-Kac-Formel skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Feynman-Kac-Formel</span></h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="contentSub">
<div id="mw-content-subtitle"></div>
</div>
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Satz von Feynman-Kac</b> ist ein Ergebnis der <a href="Wahrscheinlichkeitstheorie" title="Wahrscheinlichkeitstheorie">Wahrscheinlichkeitstheorie</a>, das z. B. in der <a href="Finanzmathematik" title="Finanzmathematik">Finanzmathematik</a> Anwendung findet. Er verbindet <a href="Stochastischer_Prozess" title="Stochastischer Prozess">stochastische Prozesse</a> aus der Wahrscheinlichkeitstheorie mit der Theorie der <a href="Partielle_Differentialgleichung" title="Partielle Differentialgleichung">partiellen</a> <a href="Differentialgleichung" title="Differentialgleichung">Differentialgleichungen</a>. Der Name geht auf <a href="Richard_Feynman" title="Richard Feynman">Richard Feynman</a> und <a href="Mark_Kac" title="Mark Kac">Mark Kac</a> zurück.
</p>
<div class="mw-heading mw-heading2"><h2 id="Aussage_des_Satzes">Aussage des Satzes</h2></div>
<p>Sei zunächst <span class="mwe-math-element mwe-math-element-inline"><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}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82120d04dfb3cbadc4912951dd12b5568c9cd8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}" loading="lazy"></span> ein an die <a href="Filtrierung_(Wahrscheinlichkeitstheorie)" title="Filtrierung (Wahrscheinlichkeitstheorie)">Filtration</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{t})_{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (F_{t})_{t}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/841207968571afb8c7c29cb593354c8711d64490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.956ex; height:2.843ex;" alt="{\displaystyle (F_{t})_{t}}" loading="lazy"></span> <a href="Adaptierter_stochastischer_Prozess" title="Adaptierter stochastischer Prozess">adaptierter Prozess</a> und Lösung der stochastischen Differentialgleichung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dX_{t}=\sigma (t,X_{t})dW_{t}+\mu (t,X_{t})dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>σ<!-- σ --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mi>d</mi>
<msub>
<mi>W</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle dX_{t}=\sigma (t,X_{t})dW_{t}+\mu (t,X_{t})dt}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8788ff75958544ef95ea04e80ca0190b77d988a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.793ex; height:2.843ex;" alt="{\displaystyle dX_{t}=\sigma (t,X_{t})dW_{t}+\mu (t,X_{t})dt}" loading="lazy"></span>.</dd></dl>
<p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{t})_{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (X_{t})_{t}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9227b1a1f8dca0ef154210b4c8b4c4f934b3706.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.385ex; height:2.843ex;" alt="{\displaystyle (X_{t})_{t}}" loading="lazy"></span> ist daher ein <a href="It%C5%8D-Prozess" class="mw-redirect" title="Itō-Prozess">Itō-Prozess</a>. Sei ferner
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:\mathbb {R} \rightarrow \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
<mo>:</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h:\mathbb {R} \rightarrow \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7132bad98312911aeb02354f0c9038ffc1704591.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.246ex; height:2.176ex;" alt="{\displaystyle h:\mathbb {R} \rightarrow \mathbb {R} }" loading="lazy"></span></dd></dl>
<p>eine beschränkte, <a href="Borel-Ma%C3%9F" class="mw-redirect" title="Borel-Maß">Borel-messbare</a> Funktion und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t,x)=\mathbb {E} (h(X_{T})\mid X_{t}=x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">E</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>∣<!-- ∣ --></mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(t,x)=\mathbb {E} (h(X_{T})\mid X_{t}=x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3772f79651e905be48e211a44606dbccc0057c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.164ex; height:2.843ex;" alt="{\displaystyle g(t,x)=\mathbb {E} (h(X_{T})\mid X_{t}=x)}" loading="lazy"></span> die an die Information in <span class="mwe-math-element mwe-math-element-inline"><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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> bedingte Erwartung ihres Wertes in <span class="mwe-math-element mwe-math-element-inline"><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}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>T</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{T}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b14f78347fa26441aac6849039d45834e6495aa0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.314ex; height:2.509ex;" alt="{\displaystyle X_{T}}" loading="lazy"></span>. Dann erfüllt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> die partielle (nicht-stochastische!) Differentialgleichung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{t}(t,x)+g_{x}(t,x)\mu (t,x)+{\frac {1}{2}}g_{xx}(t,x)\sigma (t,x)^{2}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>μ<!-- μ --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mi>x</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>σ<!-- σ --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{t}(t,x)+g_{x}(t,x)\mu (t,x)+{\frac {1}{2}}g_{xx}(t,x)\sigma (t,x)^{2}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d166b8253f1f6ebdd492c1dfbf31f75bc67764bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.227ex; height:5.176ex;" alt="{\displaystyle g_{t}(t,x)+g_{x}(t,x)\mu (t,x)+{\frac {1}{2}}g_{xx}(t,x)\sigma (t,x)^{2}=0}" loading="lazy"></span></dd></dl>
<p>mit der <a href="Randbedingung" title="Randbedingung">Randbedingung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(T,x)=h(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>T</mi>
<mo>,</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>h</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(T,x)=h(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/764429aa58ca6377990edbcf50715b39f5c88e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.502ex; height:2.843ex;" alt="{\displaystyle g(T,x)=h(x)}" loading="lazy"></span>.
</p><p>Der Beweis verwendet die <a href="Martingal" title="Martingal">Martingaleigenschaft</a> der bedingten Erwartung und die Tatsache, dass ein Itō-Prozess (gegeben in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>) genau dann Martingal ist, wenn sein Driftterm verschwindet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Zum Beispiel könnte <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle h}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> die Auszahlung eines Finanzinstruments (etwa eine <a href="Kaufoption" title="Kaufoption">Kaufoption</a>) sein, basierend auf dem Wert von <span class="mwe-math-element mwe-math-element-inline"><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}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{t}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82120d04dfb3cbadc4912951dd12b5568c9cd8f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.509ex;" alt="{\displaystyle X_{t}}" loading="lazy"></span> (etwa eine Aktie). Dann beschreibt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> den Preisprozess dieses Instruments. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{x}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{x}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58ee8ff455f045cef532e94a4801394972573cb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.282ex; height:2.009ex;" alt="{\displaystyle g_{x}}" loading="lazy"></span> ist die Ableitung des Preises vom Basiswert, im Fall einer Option ist daher <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {g_{x}}{g}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mi>g</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {g_{x}}{g}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b3c46d602a4c357dff603c8ed0789c37c557e09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.118ex; height:5.343ex;" alt="{\displaystyle {\frac {g_{x}}{g}}}" loading="lazy"></span> ihr <a href="Option_(Wirtschaft)#Delta" title="Option (Wirtschaft)">Delta</a>. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {g_{t}}{g}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mi>g</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {g_{t}}{g}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66f7c12651c1c306bdfcfdf6a9024c685762549c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.771ex; height:5.343ex;" alt="{\displaystyle {\frac {g_{t}}{g}}}" loading="lazy"></span> ist im Fall einer Kaufoption das Theta.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Bernt Øksendal: <i>Stochastic Differential Equations: An Introduction with Applications.</i> 6. Auflage, Springer, Berlin 2003, ISBN 978-3-540-04758-2.</li>
<li>John Michael Steele: <i>Stochastic Calculus and Financial Applications.</i> Springer, New York 2001, ISBN 0-387-95016-8.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-09-26" href="https://de.wikipedia.org/wiki/?title=Feynman-Kac-Formel&oldid=260086569">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
<script src="./_webp_/webpHandler.js"></script>
</body></html>