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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 Kolmogorov backward equations (diffusion) ： ウィキペディア英語版 Kolmogorov backward equations (diffusion) The Kolmogorov backward equation (KBE) (diffusion) and its adjoint sometimes known as the Kolmogorov forward equation (diffusion) are partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov processes. Both were published by Andrey Kolmogorov in 1931.〔Andrei Kolmogorov, "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" (On Analytical Methods in the Theory of Probability), 1931, ()〕 Later it was realized that the forward equation was already known to physicists under the name Fokker–Planck equation; the KBE on the other hand was new. Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state ''x'' of the system at time ''t'' (namely a probability distribution $p\_t(x)$); we want to know the probability distribution of the state at a later time $s>t$. The adjective 'forward' refers to the fact that $p\_t(x)$ serves as the initial condition and the PDE is integrated forward in time. (In the common case where the initial state is known exactly $p\_t(x)$ is a Dirac delta function centered on the known initial state). The Kolmogorov backward equation on the other hand is useful when we are interested at time ''t'' in whether at a future time ''s'' the system will be in a given subset of states ''B'', sometimes called the ''target set''. The target is described by a given function $u\_s(x)$ which is equal to 1 if state ''x'' is in the target set at time ''s'', and zero otherwise. In other words, $u\_s(x)\; =\; 1\_B$, the indicator function for the set ''B''. We want to know for every state ''x'' at time ==Formulating the Kolmogorov backward equation== Assume that the system state $x(t)$ evolves according to the stochastic differential equation :$dx(t)\; =\; \backslash mu(x(t),t)\backslash ,dt\; +\; \backslash sigma(x(t),t)\backslash ,dW(t)$ then the Kolmogorov backward equation is as follows 〔Risken, H., "The Fokker-Planck equation: Methods of solution and applications" 1996, Springer〕 :$-\backslash fracp(x,t)=\backslash mu(x,t)\backslash fracp(x,t)\; +\; \backslash frac\backslash sigma^2(x,t)\backslash frac\{\backslash partial\; x^\{2\}\}p(x,t)$ for $t\backslash le\; s$, subject to the final condition $p(x,s)=u\_s(x)$. This can be derived using Itō's lemma on $p(x,t)$ and setting the dt term equal to zero. This equation can also be derived from the Feynman-Kac formula by noting that the hit probability is the same as the expected value of $u\_s(x)$ over all paths that originate from state x at time t: :$P(X\_s\; \backslash in\; B\; \backslash mid\; X\_t\; =\; x)\; =\; E(\backslash mid\; X\_t\; =\; x)$ Historically of course the KBE 〔 was developed before the Feynman-Kac formula (1949). 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「Kolmogorov backward equations (diffusion)」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.