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<title>Itō-Formel</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Itō-Formel</h1>

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<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">Die Itō-Formel (auch Itō-<a href="Wolfgang_D%C3%B6blin" title="Wolfgang Döblin">Döblin</a>-Formel; selten auch Lemma von Itō), benannt nach dem japanischen Mathematiker <a href="It%C5%8D_Kiyoshi" title="Itō Kiyoshi">Itō Kiyoshi</a>, ist eine zentrale Aussage in der <a href="Stochastische_Analysis" title="Stochastische Analysis">stochastischen Analysis</a>. In seiner einfachsten Form ist es eine Integraldarstellung für <a href="Stochastischer_Prozess" title="Stochastischer Prozess">stochastische Prozesse</a>, die Funktionen eines <a href="Wiener-Prozess" class="mw-redirect" title="Wiener-Prozess">Wiener-Prozesses</a> sind. Es entspricht damit der Kettenregel bzw. Substitutionsregel der klassischen <a href="Differentialrechnung" title="Differentialrechnung">Differential-</a> und <a href="Integralrechnung" title="Integralrechnung">Integralrechnung</a>.

Itô publizierte 1951 einen Beweis.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Version_für_Wiener-Prozesse">Version für Wiener-Prozesse</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (W_{t})_{t\geq 0}}">

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<annotation encoding="application/x-tex">{\displaystyle (W_{t})_{t\geq 0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c15535196179ed7c35f9c2d5a869bc70ab9c7526.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.755ex; height:2.843ex;" alt="{\displaystyle (W_{t})_{t\geq 0}}" loading="lazy"> ein (Standard-)Wiener-Prozess und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aad637bebb261539130e516d148b93b069349c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.343ex; height:2.176ex;" alt="{\displaystyle h\colon \mathbb {R} \to \mathbb {R} }" loading="lazy"> eine zweimal <a href="Differentialrechnung" title="Differentialrechnung">stetig differenzierbare</a> Funktion. Dann gilt

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<annotation encoding="application/x-tex">{\displaystyle h(W_{t})=h(W_{0})+\int _{0}^{t}h'(W_{s})\,{\rm {d}}W_{s}+{\frac {1}{2}}\int _{0}^{t}h''(W_{s})\,{\rm {d}}s\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2369fb089f362ff77d9939b0034467cc4879447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.277ex; height:6.176ex;" alt="{\displaystyle h(W_{t})=h(W_{0})+\int _{0}^{t}h'(W_{s})\,{\rm {d}}W_{s}+{\frac {1}{2}}\int _{0}^{t}h''(W_{s})\,{\rm {d}}s\,.}" loading="lazy"></dd></dl>

Dabei ist das erste Integral als <a href="It%C5%8D-Integral" class="mw-redirect" title="Itō-Integral">Itō-Integral</a> und das zweite Integral als ein gewöhnliches <a href="Riemann-Integral" class="mw-redirect" title="Riemann-Integral">Riemann-Integral</a> (über die stetigen Pfade des Integranden) zu verstehen.

Für den durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{t}=h(W_{t})}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{t}=h(W_{t})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42cad4fcee2ee78ef73ed4ce9ab78b8431328441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.443ex; height:2.843ex;" alt="{\displaystyle Y_{t}=h(W_{t})}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle t\geq 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/248525429e9cd266f53ab8c52d17bc206c546060.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.101ex; height:2.343ex;" alt="{\displaystyle t\geq 0}" loading="lazy"> definierten Prozess lautet diese Darstellung in <a href="Differential_(Mathematik)" title="Differential (Mathematik)">Differentialschreibweise</a>

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<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}Y_{t}=h'(W_{t})\,{\rm {d}}W_{t}+{\frac {1}{2}}h''(W_{t})\,{\rm {d}}t\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f104e624e5c49e5eb489565d5696eaa5084a879.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.817ex; height:5.176ex;" alt="{\displaystyle {\rm {d}}Y_{t}=h'(W_{t})\,{\rm {d}}W_{t}+{\frac {1}{2}}h''(W_{t})\,{\rm {d}}t\,.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Version_für_Itō-Prozesse">Version für Itō-Prozesse</h2></div>

Ein stochastischer Prozess <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{t})_{t\geq 0}}">

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<annotation encoding="application/x-tex">{\displaystyle (X_{t})_{t\geq 0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21296940fbeeae13b27893bdb77c0ae6a02ad23f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.486ex; height:2.843ex;" alt="{\displaystyle (X_{t})_{t\geq 0}}" loading="lazy"> heißt <a href="It%C5%8D-Prozess" class="mw-redirect" title="Itō-Prozess">Itō-Prozess</a>, falls

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<annotation encoding="application/x-tex">{\displaystyle X_{t}=X_{0}+\int _{0}^{t}a_{s}\,{\rm {d}}s+\int _{0}^{t}b_{s}\,{\rm {d}}W_{s}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d871650e6c899c611ad9188a776c7857d79d6f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.743ex; height:6.176ex;" alt="{\displaystyle X_{t}=X_{0}+\int _{0}^{t}a_{s}\,{\rm {d}}s+\int _{0}^{t}b_{s}\,{\rm {d}}W_{s}}" loading="lazy"></dd></dl>

für zwei stochastische Prozesse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{s}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f0573bb76faf03c96f4f7f70f92ccba01ab0ce6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.233ex; height:2.009ex;" alt="{\displaystyle a_{s}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{s}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{s}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdabdabfe91bfe53fc624d151a46f30297afc756.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.001ex; height:2.509ex;" alt="{\displaystyle b_{s}}" loading="lazy"> gilt (genaueres dazu unter <a href="Stochastische_Integration" title="Stochastische Integration">stochastische Integration</a>). In Differentialschreibweise:

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<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}X_{t}=a_{t}\,{\rm {d}}t+b_{t}\,{\rm {d}}W_{t}\,.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/723633ba7d895c7a2fadc7b99c37d1eae98d9cac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.113ex; height:2.509ex;" alt="{\displaystyle {\rm {d}}X_{t}=a_{t}\,{\rm {d}}t+b_{t}\,{\rm {d}}W_{t}\,.}" loading="lazy"></dd></dl>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\colon \mathbb {R} _{+}\times \mathbb {R} \to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle h\colon \mathbb {R} _{+}\times \mathbb {R} \to \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce186619ce9bfb2edf0ee55e9d87cf1e8c885c03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.372ex; height:2.509ex;" alt="{\displaystyle h\colon \mathbb {R} _{+}\times \mathbb {R} \to \mathbb {R} }" loading="lazy">

eine in der ersten Komponente einmal und in der zweiten zweimal stetig differenzierbare Funktion, so ist auch der durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{t}:=h(t,X_{t})}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{t}:=h(t,X_{t})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/306cd83843fee337471da9d9a258cd24648fe21f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.694ex; height:2.843ex;" alt="{\displaystyle Y_{t}:=h(t,X_{t})}" loading="lazy"> definierte Prozess ein Itō-Prozess, und es gilt<a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\rm {d}}Y_{t}&={\frac {\partial h}{\partial t}}(t,X_{t})\,{\rm {d}}t+{\frac {\partial h}{\partial x}}(t,X_{t})\,{\rm {d}}X_{t}+{\frac {1}{2}}{\frac {\partial ^{2}h}{\partial x^{2}}}(t,X_{t})({\rm {d}}X_{t})^{2}\\&=\left({\frac {\partial h}{\partial x}}(t,X_{t})\,a_{t}+{\frac {\partial h}{\partial t}}(t,X_{t})+{\frac {1}{2}}{\frac {\partial ^{2}h}{\partial x^{2}}}(t,X_{t})\,b_{t}^{2}\right){\rm {d}}t+{\frac {\partial h}{\partial x}}(t,X_{t})\,b_{t}\,{\rm {d}}W_{t}\,.\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3d38b6a8dc757cf05b19bceb60652e38c64dbd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.314ex; margin-bottom: -0.191ex; width:77.389ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}{\rm {d}}Y_{t}&={\frac {\partial h}{\partial t}}(t,X_{t})\,{\rm {d}}t+{\frac {\partial h}{\partial x}}(t,X_{t})\,{\rm {d}}X_{t}+{\frac {1}{2}}{\frac {\partial ^{2}h}{\partial x^{2}}}(t,X_{t})({\rm {d}}X_{t})^{2}\\&=\left({\frac {\partial h}{\partial x}}(t,X_{t})\,a_{t}+{\frac {\partial h}{\partial t}}(t,X_{t})+{\frac {1}{2}}{\frac {\partial ^{2}h}{\partial x^{2}}}(t,X_{t})\,b_{t}^{2}\right){\rm {d}}t+{\frac {\partial h}{\partial x}}(t,X_{t})\,b_{t}\,{\rm {d}}W_{t}\,.\end{aligned}}}" loading="lazy"></dd></dl>

Hierbei bezeichnen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\partial h}{\partial t}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial h}{\partial t}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ae3845e7ec9761b3d7fbc6004b2a3ef4e06303e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.715ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\partial h}{\partial t}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\partial h}{\partial x}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial h}{\partial x}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b94e9b239aee6a131f660930ce19d08664e3eb24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.715ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\partial h}{\partial x}}}" loading="lazy"> die <a href="Partielle_Ableitung" title="Partielle Ableitung">partiellen Ableitungen</a> der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><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"> nach der ersten bzw. zweiten Variablen.

Die zweite Darstellung folgt aus der ersten durch Einsetzen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\rm {d}}X_{t})^{2}=b_{t}^{2}\,{\rm {d}}t}">

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<annotation encoding="application/x-tex">{\displaystyle ({\rm {d}}X_{t})^{2}=b_{t}^{2}\,{\rm {d}}t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9005ec1f0c68e8cc3b32888be790f34f8df8b103.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.576ex; height:3.343ex;" alt="{\displaystyle ({\rm {d}}X_{t})^{2}=b_{t}^{2}\,{\rm {d}}t}" loading="lazy"> und Zusammenfassen der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {d}}t}">

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<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb64fd51c6f515ec3c7adf73e0ac70bcac78c8f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle {\rm {d}}t}" loading="lazy">- und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {d}}W_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}W_{t}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42c26ef0168c9b5b20465bbd17e1927ef56e12b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.312ex; height:2.509ex;" alt="{\displaystyle {\rm {d}}W_{t}}" loading="lazy">-Terme.

<div class="mw-heading mw-heading3"><h3 id="Mehrdimensionale_Version">Mehrdimensionale Version</h3></div>

Die Formel lässt sich auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" 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}" loading="lazy"> Itō-Prozesse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=(X_{1},\dots ,X_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle X=(X_{1},\dots ,X_{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8748da867281077ef31c4b5120f727043ce05943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.187ex; height:2.843ex;" alt="{\displaystyle X=(X_{1},\dots ,X_{n})}" loading="lazy"> verallgemeinern. Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:[0,\infty )\times \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle h:[0,\infty )\times \mathbb {R} ^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/288328b219da652a2f8bc9a74fad84737f16c380.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.085ex; height:2.843ex;" alt="{\displaystyle h:[0,\infty )\times \mathbb {R} ^{n}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}">

<msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd24bae0d7570018e828e19851902c09c618af91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}" loading="lazy"> in der ersten und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle C^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fd6a5946b7e916352b0afc557f992328bac85e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{2}}" loading="lazy"> in den restlichen Variablen. Definiere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(t):=h(t,X(t))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y(t):=h(t,X(t))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d229d2b87b142424bc8467a6fe3cf8bfa5979b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.818ex; height:2.843ex;" alt="{\displaystyle Y(t):=h(t,X(t))}" loading="lazy"> dann gilt

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}Y(t)={\frac {\partial h}{\partial t}}(t,X(t))\,{\rm {d}}t+\sum \limits _{i=1}^{n}{\frac {\partial h}{\partial i}}(t,X(t))\,{\rm {d}}X_{i}(t)+{\frac {1}{2}}\sum \limits _{i,j=1}^{n}{\frac {\partial ^{2}h}{\partial i\partial j}}(t,X(t)){\rm {d}}[X_{i},X_{j}](t).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fd827aa1e55b160692bf22b26cfc993bee3a6ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:84.012ex; height:7.176ex;" alt="{\displaystyle {\rm {d}}Y(t)={\frac {\partial h}{\partial t}}(t,X(t))\,{\rm {d}}t+\sum \limits _{i=1}^{n}{\frac {\partial h}{\partial i}}(t,X(t))\,{\rm {d}}X_{i}(t)+{\frac {1}{2}}\sum \limits _{i,j=1}^{n}{\frac {\partial ^{2}h}{\partial i\partial j}}(t,X(t)){\rm {d}}[X_{i},X_{j}](t).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Version_für_Semimartingale">Version für Semimartingale</h2></div>

<msub>

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</msub>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle (X_{t})_{t\geq 0}=(X_{t}^{1},\dotsc ,X_{t}^{d})_{t\geq 0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eff2f58940859f98c56006952b5d3b224f69ea72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.638ex; height:3.176ex;" alt="{\displaystyle (X_{t})_{t\geq 0}=(X_{t}^{1},\dotsc ,X_{t}^{d})_{t\geq 0}}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{d}}">

<msup>

</mrow>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{d}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a713426956296f1668fce772df3c60b9dde8a685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.77ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{d}}" loading="lazy">-wertiges <a href="Semimartingal" title="Semimartingal">Semimartingal</a> und sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\in C^{2}(\mathbb {R} ^{d},\mathbb {R} )}">

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle F\in C^{2}(\mathbb {R} ^{d},\mathbb {R} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f26293ec50f76412d3a5850c947b18637335074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.725ex; height:3.176ex;" alt="{\displaystyle F\in C^{2}(\mathbb {R} ^{d},\mathbb {R} )}" loading="lazy">. Dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F(X_{t}))_{t\geq 0}}">

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle (F(X_{t}))_{t\geq 0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/928f0bcf807e453ddd4f9529a90f1b67a2ff9da9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.036ex; height:2.843ex;" alt="{\displaystyle (F(X_{t}))_{t\geq 0}}" loading="lazy"> wieder ein Semimartingal und es gilt

<mtr>

<mtd>

<msub>

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<msub>

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</msub>

</mtd>

<mtd>

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<mfrac>

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<mrow>

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<mfrac>

<mrow>

<msup>

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</msup>

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<mrow>

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</msup>

<msup>

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</msup>

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<msub>

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<msup>

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</msup>

<msup>

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</msup>

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</mtr>

<mtr>

<mtd>

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</munder>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

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</msup>

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</mfrac>

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<msub>

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</msub>

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</mtr>

</mtable>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(X_{t})-F(X_{0})=&\sum _{j=1}^{d}\int _{0+}^{t}{\frac {\partial F}{\partial x^{j}}}(X_{s-}){\rm {d}}X_{s}^{j}+{\frac {1}{2}}\sum _{j,k=1}^{d}\int _{0+}^{t}{\frac {\partial ^{2}F}{\partial x^{j}\partial x^{k}}}(X_{s-}){\rm {d}}[X^{j},X^{k}]_{s}^{c}\\&{}+\sum _{0<s\leq t}\left(F(X_{s})-F(X_{s-})-\sum _{j=1}^{d}{\frac {\partial F}{\partial x^{j}}}(X_{s-})\Delta X_{s}^{j}\right).\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23758421b1b1cf32a6c54a4ecdb59230a212c6ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:80.123ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}F(X_{t})-F(X_{0})=&\sum _{j=1}^{d}\int _{0+}^{t}{\frac {\partial F}{\partial x^{j}}}(X_{s-}){\rm {d}}X_{s}^{j}+{\frac {1}{2}}\sum _{j,k=1}^{d}\int _{0+}^{t}{\frac {\partial ^{2}F}{\partial x^{j}\partial x^{k}}}(X_{s-}){\rm {d}}[X^{j},X^{k}]_{s}^{c}\\&{}+\sum _{0<s\leq t}\left(F(X_{s})-F(X_{s-})-\sum _{j=1}^{d}{\frac {\partial F}{\partial x^{j}}}(X_{s-})\Delta X_{s}^{j}\right).\end{aligned}}}" loading="lazy"></dd></dl>

Hierbei bedeutet:

<msub>

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</msub>

</mrow>

</munder>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle X_{s-}=\lim _{u\uparrow s}X_{u}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20bc36e8000c52906677b0f7be5fee658ebcb1a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.784ex; height:2.843ex;" alt="{\displaystyle \textstyle X_{s-}=\lim _{u\uparrow s}X_{u}}" loading="lazy"> der linksseitige Grenzwert,</li>

<li>das Integrationsgebiet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{[0+,t]}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{[0+,t]}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa37f15af3f93df5391db786346bc8d7485dc6ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.461ex; height:3.009ex;" alt="{\displaystyle 1_{[0+,t]}}" loading="lazy"> bedeutet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{(0,t]}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{(0,t]}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00eeb8b9a89f59cb17dbbaf9d29bffd678ef65cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.365ex; height:3.009ex;" alt="{\displaystyle 1_{(0,t]}}" loading="lazy">. Ein Semimartingal kann bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle X_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6381fdad2b9f11954b1fc2db08bbaccf634ededa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{0}}" loading="lazy"> einen Sprung haben, das heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}\neq X_{0+}}">

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle X_{0}\neq X_{0+}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bc761c94f0644f307d8c7445e0d113876e7554f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.334ex; height:2.676ex;" alt="{\displaystyle X_{0}\neq X_{0+}}" loading="lazy"> und somit wird sichergestellt, dass nur über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,t]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,t]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49ec684b68e2842f5288a7ea9d08e8ec2b1ea704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.588ex; height:2.843ex;" alt="{\displaystyle (0,t]}" loading="lazy"> integriert wird und der Anfangswert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(X_{0})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle F(X_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/246da967fb11e8faea37eaa8a0cdf505ce791b31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.529ex; height:2.843ex;" alt="{\displaystyle F(X_{0})}" loading="lazy"> wird deshalb nicht über das Integral gedeckt.</li>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \Delta X_{s}^{j}=X_{s}^{j}-X_{s-}^{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/366abce23517b8ebab601000fb3109def5ff2ff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.936ex; height:3.509ex;" alt="{\displaystyle \Delta X_{s}^{j}=X_{s}^{j}-X_{s-}^{j}}" loading="lazy"> der zugehörige <a href="Sprungprozess" title="Sprungprozess">Sprungprozess</a>.</li>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle [X^{j},X^{k}]^{c}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfb3280b1de91dcf4fb6073f0b969d122c09a4a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.264ex; height:3.176ex;" alt="{\displaystyle [X^{j},X^{k}]^{c}}" loading="lazy"> wird die <a href="Variation_(Mathematik)#Quadratische_Variation" title="Variation (Mathematik)">quadratische Kovariation</a> der stetigen Anteile der Komponenten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{j}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle X^{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b93dfcd69ade2805dac4937260e66b1d3787194d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.907ex; height:2.676ex;" alt="{\displaystyle X^{j}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{k}}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26829a7c0c84cd8a07f32ae14897a39dffb8f577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.086ex; height:2.676ex;" alt="{\displaystyle X^{k}}" loading="lazy"> bezeichnet.</li></ul>

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> ein stetiges Semimartingal ist, verschwindet die letzte große Klammer nach dem Plus und es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X^{j},X^{k}]^{c}=[X^{j},X^{k}]}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle [X^{j},X^{k}]^{c}=[X^{j},X^{k}]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2a43b9f05a2932d1e7c431e8037ff81d6088d25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.682ex; height:3.176ex;" alt="{\displaystyle [X^{j},X^{k}]^{c}=[X^{j},X^{k}]}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Bemerkung">Bemerkung</h3></div>

Schreibt man den Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X^{j},X^{k}]_{t}^{c}:=[X^{j},X^{k}]_{t}-\sum \limits _{s\leq t}\Delta X_{s}^{j}\Delta X_{s}^{k}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle [X^{j},X^{k}]_{t}^{c}:=[X^{j},X^{k}]_{t}-\sum \limits _{s\leq t}\Delta X_{s}^{j}\Delta X_{s}^{k}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b7e256d7be97ec943487fec9fe4822ef1f52ea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.622ex; height:5.676ex;" alt="{\displaystyle [X^{j},X^{k}]_{t}^{c}:=[X^{j},X^{k}]_{t}-\sum \limits _{s\leq t}\Delta X_{s}^{j}\Delta X_{s}^{k}}" loading="lazy"> aus, so erhält man für eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C^{2}(\mathbb {R} ^{d},\mathbb {R} )}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f\in C^{2}(\mathbb {R} ^{d},\mathbb {R} )}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(X_{t})-f(X_{0})=&\sum _{j=1}^{d}\int _{0+}^{t}{\frac {\partial f}{\partial x^{j}}}(X_{s-}){\rm {d}}X_{s}^{j}+{\frac {1}{2}}\sum _{j,k=1}^{d}\int _{0+}^{t}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}(X_{s-}){\rm {d}}[X^{j},X^{k}]_{s}\\&{}+\sum _{0<s\leq t}\left(\Delta f(X_{s})-\sum _{j=1}^{d}{\frac {\partial f}{\partial x^{j}}}(X_{s-})\Delta X_{s}^{j}-{\frac {1}{2}}\sum _{k,j=1}^{d}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}(X_{s-})\Delta X_{s}^{j}\Delta X_{s}^{k}\right).\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f1acdfc2a743bfc0881687eac0e0c7741e4af7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:94.178ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}f(X_{t})-f(X_{0})=&\sum _{j=1}^{d}\int _{0+}^{t}{\frac {\partial f}{\partial x^{j}}}(X_{s-}){\rm {d}}X_{s}^{j}+{\frac {1}{2}}\sum _{j,k=1}^{d}\int _{0+}^{t}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}(X_{s-}){\rm {d}}[X^{j},X^{k}]_{s}\\&{}+\sum _{0<s\leq t}\left(\Delta f(X_{s})-\sum _{j=1}^{d}{\frac {\partial f}{\partial x^{j}}}(X_{s-})\Delta X_{s}^{j}-{\frac {1}{2}}\sum _{k,j=1}^{d}{\frac {\partial ^{2}f}{\partial x^{j}\partial x^{k}}}(X_{s-})\Delta X_{s}^{j}\Delta X_{s}^{k}\right).\end{aligned}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f(X_{s}):=f(X_{s})-f(X_{s-})}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta f(X_{s}):=f(X_{s})-f(X_{s-})}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Für_das_Stratonowitsch-Integral">Für das Stratonowitsch-Integral</h3></div>

<div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="Stratonowitsch-Integral#Itō-Formeln" title="Stratonowitsch-Integral">Stratonowitsch-Integral#Itō-Formeln</a></div>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">-Semimartingal und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C^{2}(\mathbb {R} ^{n},\mathbb {R} )}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d60325026e406ae7870aef8731dc5bb8b0e5f1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.389ex; height:3.176ex;" alt="{\displaystyle f\in C^{2}(\mathbb {R} ^{n},\mathbb {R} )}" loading="lazy">, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b884e2d65b3356219702968b6751485fb8f38570.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle f(X)}" loading="lazy"> ein Semimartingal und es gilt<a href="#cite_note-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle f(X_{t})-f(X_{0})=\sum \limits _{i=1}^{n}\int _{0+}^{t}{\frac {\partial f}{\partial x_{i}}}(X_{s-})\circ dX_{s}^{i}+\sum \limits _{0<s\leq t}\left(f(X_{s})-f(X_{s-})-\sum \limits _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(X_{s-})\Delta X_{s}^{i}\right).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5231d693a84082d3120f041ac074bea236ced9da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:92.816ex; height:7.509ex;" alt="{\displaystyle f(X_{t})-f(X_{0})=\sum \limits _{i=1}^{n}\int _{0+}^{t}{\frac {\partial f}{\partial x_{i}}}(X_{s-})\circ dX_{s}^{i}+\sum \limits _{0<s\leq t}\left(f(X_{s})-f(X_{s-})-\sum \limits _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(X_{s-})\Delta X_{s}^{i}\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Version_für_Funktionen_mit_beschränkter_quadratischer_Variation">Version für Funktionen mit beschränkter quadratischer Variation</h2></div>

<a href="Hans_F%C3%B6llmer" title="Hans Föllmer">Hans Föllmer</a> erweiterte die Formel von Itō auf (deterministische) Funktionen mit beschränkter quadratischer Variation.<a href="#cite_note-4">[4]</a>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in C^{2}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd393b6aadadbc7168963960a0aec315d4cb8da2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.971ex; height:3.009ex;" alt="{\displaystyle f\in C^{2}}" loading="lazy"> eine reell-wertige Funktion und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:{[0,\infty [}\to \mathbb {R} }">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4bce9594d5d76854b0711eab5482f6147da67360.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.373ex; height:2.843ex;" alt="{\displaystyle x:{[0,\infty [}\to \mathbb {R} }" loading="lazy"> eine <a href="C%C3%A0dl%C3%A0g-Funktion" title="Càdlàg-Funktion">Càdlàg-Funktion</a> mit endlicher quadratischer Variation. Dann gilt

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x_{t})&=f(x_{0})+\int _{0}^{t}f'(x_{s-})\mathrm {d} x_{s}+{\frac {1}{2}}\int _{]0,t]}f''(x_{s-})d[x,x]_{s}\\&+\sum _{0\leq s\leq t}\left(f(x_{s})-f(x_{s-})-f'(x_{s-})\Delta x_{s}-{\frac {1}{2}}f''(x_{s-})(\Delta x_{s})^{2})\right).\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2c66519e0ead3a3187aacf652d264607741b8e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.976ex; margin-bottom: -0.195ex; width:68.672ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}f(x_{t})&=f(x_{0})+\int _{0}^{t}f'(x_{s-})\mathrm {d} x_{s}+{\frac {1}{2}}\int _{]0,t]}f''(x_{s-})d[x,x]_{s}\\&+\sum _{0\leq s\leq t}\left(f(x_{s})-f(x_{s-})-f'(x_{s-})\Delta x_{s}-{\frac {1}{2}}f''(x_{s-})(\Delta x_{s})^{2})\right).\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle Y_{t}=\sin(W_{t})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7982361fdc2eaf4087b5944b4de67191cfa9e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.96ex; height:2.843ex;" alt="{\displaystyle Y_{t}=\sin(W_{t})}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\rm {d}}Y_{t}=\cos(W_{t})\,{\rm {d}}W_{t}-{\tfrac {1}{2}}\sin(W_{t})\,{\rm {d}}t}">

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<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}Y_{t}=\cos(W_{t})\,{\rm {d}}W_{t}-{\tfrac {1}{2}}\sin(W_{t})\,{\rm {d}}t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7737e7db16b2d6e499a32f8a861348c2385ecc1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:34.296ex; height:3.509ex;" alt="{\displaystyle {\rm {d}}Y_{t}=\cos(W_{t})\,{\rm {d}}W_{t}-{\tfrac {1}{2}}\sin(W_{t})\,{\rm {d}}t}" loading="lazy">.</li></ul>

<ul><li>Mit Hilfe der Formel kann man einfach beweisen, dass die <a href="Geometrische_brownsche_Bewegung" title="Geometrische brownsche Bewegung">geometrische brownsche Bewegung</a></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle S_{t}=S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64e4b6c5ee70c5a6cecc0f5c0c64adac18f08edd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.044ex; height:3.843ex;" alt="{\displaystyle S_{t}=S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}}" loading="lazy"></dd></dl></dd>

<dd>eine Lösung der <a href="Stochastische_Differentialgleichung" title="Stochastische Differentialgleichung">stochastischen Differentialgleichung</a> von <a href="Black-Scholes-Modell" title="Black-Scholes-Modell">Black und Scholes</a>

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<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}S_{t}=rS_{t}\,{\rm {d}}t+\sigma S_{t}\,{\rm {d}}W_{t}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a376a0fb836a1d29e612e20aca6f56261233ee7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.581ex; height:2.509ex;" alt="{\displaystyle {\rm {d}}S_{t}=rS_{t}\,{\rm {d}}t+\sigma S_{t}\,{\rm {d}}W_{t}}" loading="lazy"></dd></dl></dd>

<dl><dd>Hierzu wählt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}=W_{t}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d47f40a735611cadd3595aad544ae32be31b5b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.868ex; height:2.509ex;" alt="{\displaystyle X_{t}=W_{t}}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{t}=0,\;b_{t}=1}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d4c4948d3a48d904b1bf1d59f77161c8395fc2e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.08ex; height:2.509ex;" alt="{\displaystyle a_{t}=0,\;b_{t}=1}" loading="lazy">.</dd></dl>

<dl><dd>Dann ergibt die Formel mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t,x)=S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma x}}">

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<annotation encoding="application/x-tex">{\displaystyle h(t,x)=S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma x}}</annotation>

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</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {d}}S_{t}=\left[\left(r-{\frac {\sigma ^{2}}{2}}+{\frac {\sigma ^{2}}{2}}\right)S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}\right]{\rm {d}}t+\left[\sigma S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}\right]{\rm {d}}W_{t}=rS_{t}\,{\rm {d}}t+\sigma S_{t}\,{\rm {d}}W_{t}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4361106c796478db21b77b0f13fbab02bf7939a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:90.38ex; height:6.343ex;" alt="{\displaystyle {\rm {d}}S_{t}=\left[\left(r-{\frac {\sigma ^{2}}{2}}+{\frac {\sigma ^{2}}{2}}\right)S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}\right]{\rm {d}}t+\left[\sigma S_{0}e^{rt-{\frac {1}{2}}\sigma ^{2}t+\sigma W_{t}}\right]{\rm {d}}W_{t}=rS_{t}\,{\rm {d}}t+\sigma S_{t}\,{\rm {d}}W_{t}\,.}" loading="lazy"></dd></dl></dd></dl>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbf {W} _{t})_{t\geq 0}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d4a38cf151b0d8503038eacd48886bb5d0c27cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.324ex; height:2.843ex;" alt="{\displaystyle (\mathbf {W} _{t})_{t\geq 0}}" loading="lazy"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">-dimensionaler Wiener-Prozess und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\colon \mathbb {R} ^{d}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\colon \mathbb {R} ^{d}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/358ef7046db2f0c73f611d0cefafde20496853f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.837ex; height:2.676ex;" alt="{\displaystyle F\colon \mathbb {R} ^{d}\to \mathbb {R} }" loading="lazy"> zweimal stetig differenzierbar, dann gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{t}=F(\mathbf {W} _{t})}">

<msub>

</mrow>

</msub>

<msub>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y_{t}=F(\mathbf {W} _{t})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e19ea43aa105e4a80ca5ba38bf3bba68b4db23d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.414ex; height:2.843ex;" alt="{\displaystyle Y_{t}=F(\mathbf {W} _{t})}" loading="lazy"></li></ul>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} Y_{t}=\nabla F(\mathbf {W} _{t})^{\mathsf {T}}\cdot \mathrm {d} \mathbf {W} _{t}+{\frac {1}{2}}\Delta F(\mathbf {W} _{t})\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b91a45b181ef646e0ae162d199c0f4864d962af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.987ex; height:5.176ex;" alt="{\displaystyle \mathrm {d} Y_{t}=\nabla F(\mathbf {W} _{t})^{\mathsf {T}}\cdot \mathrm {d} \mathbf {W} _{t}+{\frac {1}{2}}\Delta F(\mathbf {W} _{t})\,\mathrm {d} t}" loading="lazy">,</dd></dl></dd>

<dd>wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a16aca383ee9a3da73ee2099c3aaab7922468966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.677ex; height:2.176ex;" alt="{\displaystyle \nabla F}" loading="lazy"> den <a href="Gradient_(Mathematik)" title="Gradient (Mathematik)">Gradienten</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcde5b83b29cc20f808fb4f349b838b82ed99a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.677ex; height:2.176ex;" alt="{\displaystyle \Delta F}" loading="lazy"> den <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operator</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> bezeichnen.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Unendlich-dimensionale_Itō-Formeln">Unendlich-dimensionale Itō-Formeln</h2></div>

Es gibt verschiedene Varianten von Itō-Formeln für unendlich-dimensionale Räume (z. B. Pardoux<a href="#cite_note-5">[5]</a>, Gyöngy-Krylow<a href="#cite_note-6">[6]</a>, Brzezniak-van Neerven-Veraar-Weis<a href="#cite_note-7">[7]</a>).

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Euler-Maruyama-Verfahren" title="Euler-Maruyama-Verfahren">Euler-Maruyama-Verfahren</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Philip E. Protter: Stochastic Integration and Differential Equations (2nd edition), Springer, 2004, ISBN 3-540-00313-4.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Kiyoshi Itô: <cite style="font-style:italic">On a formula concerning stochastic differentials</cite>. In: <cite style="font-style:italic">Nagoya Math. J.</cite> Band 3, 1951, S. 55–65 (<a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/nagoya-mathematical-journal/volume-3/issue-none/On-a-formula-concerning-stochastic-differentials/nmj/1118799221.full">projecteuclid.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:It%C5%8D-Formel&rft.atitle=On+a+formula+concerning+stochastic+differentials&rft.au=Kiyoshi+It%C3%B4&rft.btitle=Nagoya+Math.+J.&rft.date=1951&rft.genre=book&rft.pages=55-65&rft.volume=3" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Hui-Hsiung Kuo: Introduction to Stochastic Integration. Springer, 2006, ISBN 978-0387-28720-1, S. 103 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=VEAxuzpCvj0C&pg=PA103#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Philip E. Protter: <cite style="font-style:italic">Stochastic Integration and Differential Equations</cite>. Hrsg.: Springer. 2004, ISBN 3-540-00313-4, S. 277–278.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:It%C5%8D-Formel&rft.au=Philip+E.+Protter&rft.btitle=Stochastic+Integration+and+Differential+Equations&rft.date=2004&rft.genre=book&rft.isbn=3540003134&rft.pages=277-278" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Hans Föllmer: <cite style="font-style:italic">Calcul d'Ito sans probabilités</cite>. In: <cite style="font-style:italic">Séminaire de probabilités de Strasbourg</cite>. Band 15, 1981, S. 143–144 (<a rel="nofollow" class="external text" href="https://www.numdam.org/item/SPS_1981__15__143_0/">numdam.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:It%C5%8D-Formel&rft.atitle=Calcul+d%27Ito+sans+probabilit%C3%A9s&rft.au=Hans+F%C3%B6llmer&rft.btitle=S%C3%A9minaire+de+probabilit%C3%A9s+de+Strasbourg&rft.date=1981&rft.genre=book&rft.pages=143-144&rft.volume=15" style="display:none">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> E. Pardoux, E: <cite style="font-style:italic">Équations aux dérivées partielles stochastiques de type monotone</cite>. In: <cite style="font-style:italic">Séminaire Jean Leray</cite>. Nr. 3, 1974 (<a rel="nofollow" class="external text" href="https://www.numdam.org/item/SJL_1974-1975___3_A2_0/">numdam.org</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:It%C5%8D-Formel&rft.atitle=%C3%89quations+aux+d%C3%A9riv%C3%A9es+partielles+stochastiques+de+type+monotone&rft.au=E.+Pardoux%2C+E&rft.date=1974&rft.genre=journal&rft.issue=3&rft.jtitle=S%C3%A9minaire+Jean+Leray" style="display:none">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> I. Gyöngy und N. V. Krylov: <cite style="font-style:italic">Ito formula in banach spaces</cite>. In: Springer, Berlin, Heidelberg (Hrsg.): <cite style="font-style:italic">Arató, M., Vermes, D., Balakrishnan, A.V. (eds) Stochastic Differential Systems</cite>. Band 36, 1981, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007/BFb0006409">10.1007/BFb0006409</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:It%C5%8D-Formel&rft.atitle=Ito+formula+in+banach+spaces&rft.au=I.+Gy%C3%B6ngy+und+N.+V.+Krylov&rft.btitle=Arat%C3%B3%2C+M.%2C+Vermes%2C+D.%2C+Balakrishnan%2C+A.V.+%28eds%29+Stochastic+Differential+Systems&rft.date=1981&rft.doi=10.1007%2FBFb0006409&rft.genre=book&rft.volume=36" style="display:none">

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Z. Brzezniak, J. M. A. M. van Neerven, M. C. Veraar und L. Weis: <cite style="font-style:italic">Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation</cite>. 2008.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:It%C5%8D-Formel&rft.au=Z.+Brzezniak%2C+J.+M.+A.+M.+van+Neerven%2C+M.+C.+Veraar+und+L.+Weis&rft.btitle=Ito%27s+formula+in+UMD+Banach+spaces+and+regularity+of+solutions+of+the+Zakai+equation&rft.date=2008&rft.genre=book" style="display:none">

</li>

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