Bougerol恒等式在法律中的推广及相应的预期路径变换
Extensions of Bougerol's identity in law and the associated anticipative path transformations
摘要：
设$B=\{ B_{t}\} _{t\ge 0}$是一维标准布朗运动，用$A_{t},\,t\ge 0$表示几何布朗运动$e^{B_{t}},\,t\ge 0$的二次变分. Bougerol的著名恒等式（1983）断言，如果$β=\{ β(t)\} _{t\ge 0}$是另一个与$B$无关的布朗运动，那么对于每个固定的$t>0$，$β(A_{t})$与$\sinh B_{t}$在定律上是相同的。本文将Bougerol恒等式推广为一个关于时间过程的恒等式$t$，它表现出布朗运动定律的某种不变性.扩展描述的$B$涉及$A_{t}$作为一个预期因素的预期变换。这些变换的Girsanov型公式。布热罗尔的身份的一个变种的扩展也提出。
英文摘要：
Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion and denote by $A_{t},\,t\ge 0$, the quadratic variation of the geometric Brownian motion $e^{B_{t}},\,t\ge 0$. Bougerol's celebrated identity (1983) asserts that, if $β=\{ β(t)\} _{t\ge 0}$ is another Brownian motion independent of $B$, then $β(A_{t})$ is identical in law with $\sinh B_{t}$ for every fixed $t>0$. In this paper, we extend Bougerol's identity to an identity in law for processes up to time $t$, which exhibits a certain invariance of the law of Brownian motion. The extension is described in terms of anticipative transforms of $B$ involving $A_{t}$ as an anticipating factor. A Girsanov-type formula for those transforms is shown. An extension of a variant of Bougerol's identity is also presented.
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