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

Abstract: 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 $\beta ={ \beta (t)} {t\ge 0}$ is another Brownian motion independent 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 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.