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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 Doléans-Dade exponential ： ウィキペディア英語版 Doléans-Dade exponential In stochastic calculus, the Doléans-Dade exponential, Doléans exponential, or stochastic exponential, of a semimartingale ''X'' is defined to be the solution to the stochastic differential equation with initial condition . The concept is named after Catherine Doléans-Dade. It is sometimes denoted by ''(unicode:Ɛ)''(''X''). In the case where ''X'' is differentiable, then ''Y'' is given by the differential equation to which the solution is . Alternatively, if for a Brownian motion ''B'', then the Doléans-Dade exponential is a geometric Brownian motion. For any continuous semimartingale ''X'', applying Itō's lemma with gives :$$ \begin d\log(Y) &= \frac\,dY -\frac\,d() \\ &= dX - \frac\,d(). \end Exponentiating gives the solution :$Y\_t\; =\; \backslash exp\backslash Bigl(X\_t-X\_0-\backslash frac12()\_t\backslash Bigr),\backslash qquad\; t\backslash ge0.$ This differs from what might be expected by comparison with the case where ''X'' is differentiable due to the existence of the quadratic variation term () in the solution. The Doléans-Dade exponential is useful in the case when ''X'' is a local martingale. Then, ''(unicode:Ɛ)''(''X'') will also be a local martingale whereas the normal exponential exp(''X'') is not. This is used in the Girsanov theorem. Criteria for a continuous local martingale ''X'' to ensure that its stochastic exponential ''(unicode:Ɛ)''(''X'') is actually a martingale are given by Kazamaki's condition, Novikov's condition, and Beneš' condition. It is possible to apply Itō's lemma for non-continuous semimartingales in a similar way to show that the Doléans-Dade exponential of any semimartingale ''X'' is :$Y\_t\; =\; \backslash exp\backslash Bigl(X\_t-X\_0-\backslash frac12()\_t\backslash Bigr)\backslash prod\_(1+\backslash Delta\; X\_s)\; \backslash exp\; \backslash Bigl(-\backslash Delta\; X\_s+\backslash frac12\backslash Delta\; X\_s^2\backslash Bigr),\backslash qquad\; t\backslash ge0,$ where the product extents over the (countable many) jumps of ''X'' up to time ''t''. == See also == * Stochastic logarithm 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「Doléans-Dade exponential」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.