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Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motionand denote by $A_{t},\,t\ge 0$, the quadratic variation of the geometricBrownian motion $e^{B_{t}},\,t\ge 0$. Bougerol's celebrated identity (1983)asserts that, if $\beta =\{ \beta (t)\} _{t\ge 0}$ is another Brownian motionindependent of $B$, then $\beta (A_{t})$ is identical in law with $\sinh B_{t}$for every fixed $t>0$. In this paper, we extend Bougerol's identity to anidentity in law for processes up to time $t$, which exhibits a certaininvariance of the law of Brownian motion. The extension is described in termsof anticipative transforms of $B$ involving $A_{t}$ as an anticipating factor.A Girsanov-type formula for those transforms is shown. An extension of avariant of Bougerol's identity is also presented.

Extensions of Bougerol's identity in law and the associated anticipative path transformations