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

TL;DR

This paper extends Bougerol's identity to process-level laws involving anticipative transformations of Brownian motion, revealing invariance properties and providing Girsanov-type formulas.

Contribution

It introduces a process-level extension of Bougerol's identity using anticipative path transformations and establishes related Girsanov formulas.

Findings

Extended Bougerol's identity to processes up to time t

Derived Girsanov-type formulas for anticipative transforms

Presented variants of Bougerol's identity with new invariance properties

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.