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.

28 pages

References in corpus (3)

• Exponential functionals of Brownian motion, I: Probability laws at fixed time
• Exponential functionals of Brownian motion, II: Some related diffusion processes
• On some identities in law involving exponential functionals of Brownian motion and Cauchy variable