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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Feynman–Kac formula ： ウィキペディア英語版 Feynman–Kac formula The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. It offers a method of solving certain PDEs by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods. Consider the PDE :$\backslash frac(x,t)\; +\; \backslash mu(x,t)\; \backslash frac(x,t)\; +\; \backslash tfrac\; \backslash sigma^2(x,t)\; \backslash frac(x,t)\; -V(x,t)\; u(x,t)\; +\; f(x,t)\; =\; 0,$ defined for all ''x'' in R and ''t'' in (''T'' ), subject to the terminal condition :$u(x,T)=\backslash psi(x),$ where μ, σ, ψ, ''V'', ''f'' are known functions, ''T'' is a parameter and $u:\backslash mathbb\backslash times()\backslash to\backslash mathbb$ is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a conditional expectation :$u(x,t)\; =\; E^Q\backslash left(\backslash int\_t^T\; e^f(X\_r,r)dr\; +\; e^\backslash psi(X\_T)\; \backslash Bigg|\; X\_t=x\; \backslash right)$ under the probability measure Q such that ''X'' is an Itō process driven by the equation :$dX\; =\; \backslash mu(X,t)\backslash ,dt\; +\; \backslash sigma(X,t)\backslash ,dW^Q,$ with ''WQ''(''t'') is a Wiener process (also called Brownian motion) under ''Q'', and the initial condition for ''X''(''t'') is ''X''(t) = ''x''. == Proof == Let ''u''(''x'', ''t'') be the solution to above PDE. Applying Itō's lemma to the process :$Y(s)\; =\; e^\; u(X\_s,s)+\; \backslash int\_t^s\; e^f(X\_r,r)\; \backslash ,\; dr$ one gets :$$ \begin dY = \,du(X_s,s) \\() & f(X_r,r) \, dr\right) \end Since :$d\backslash left(e^\backslash right)\; =-V(X\_s,s)\; e^\; \backslash ,ds,$ the third term is $O(dt\; \backslash ,\; du)$ and can be dropped. We also have that :$d\backslash left(\backslash int\_t^s\; e^f(X\_r,r)dr\backslash right)\; =\; e^\; f(X\_s,s)\; ds.$ Applying Itō's lemma once again to $du(X\_s,s)$, it follows that :$$ \begin dY= +\frac+\tfrac\sigma^2(X_s,s)\frac\right)\,ds \\() & \,dW. \end The first term contains, in parentheses, the above PDE and is therefore zero. What remains is :$dY=e^\backslash sigma(X,s)\backslash frac\backslash ,dW.$ Integrating this equation from ''t'' to ''T'', one concludes that :$Y(T)\; -\; Y(t)\; =\; \backslash int\_t^T\; e^\backslash sigma(X,s)\backslash frac\backslash ,dW.$ Upon taking expectations, conditioned on ''Xt'' = ''x'', and observing that the right side is an Itō integral, which has expectation zero, it follows that :$E(X\_t=x)\; =\; E(X\_t=x)\; =\; u(x,t).$ The desired result is obtained by observing that : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Feynman–Kac formula」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.