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In mathematics, Itô's lemma is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. Typically, it is memorized by forming the Taylor series expansion of the function up to its second derivatives and identifying the square of an increment in the Wiener process with an increment in time. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values. Itô's lemma, which is named after Kiyosi Itô, is occasionally referred to as the Itô–Doeblin theorem in recognition of the recently discovered work of Wolfgang Doeblin.〔("Stochastic Calculus :: Itô–Doeblin formula", Michael Stastny )〕 Note that while Ito's lemma was proved by Kiyosi Itô, Itô's theorem, a result in group theory, is due to Noboru Itô.〔(Ito's Lemma - from Wolfram MathWorld. )〕 == Informal derivation == A formal proof of the lemma relies on taking the limit of a sequence of random variables. This approach is not presented here since it involves a number of technical details. Instead, we give a sketch of how one can derive Itô's lemma by expanding a Taylor series and applying the rules of stochastic calculus. Assume is a Itô drift-diffusion process that satisfies the stochastic differential equation : where is a Wiener process. If is a twice-differentiable scalar function, its expansion in a Taylor series is : Substituting for and for gives : In the limit as , the terms and tend to zero faster than , which is . Setting the and terms to zero, substituting for , and collecting the and terms, we obtain : as required. 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「Itô's lemma」の詳細全文を読む スポンサード リンク
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