  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.
