A Girsanov-type formula for a class of anticipative transforms of Brownian motion associated with exponential functionals

TL;DR

This paper establishes a Girsanov-type formula for anticipative transforms of Brownian motion involving exponential functionals, unifying previous results and exploring invariance properties under perturbations.

Contribution

It introduces a unified Girsanov-type formula for anticipative Brownian motion transforms with exponential functionals, extending prior work and analyzing invariance under perturbations.

Findings

Unified Girsanov-type formula for anticipative transforms

Invariance of Brownian law under certain anticipative perturbations

Disintegration formula for Wiener measure related to exponential functionals

Abstract

In this paper, with the help of a result by Matsumoto--Yor (2000), we prove a Girsanov-type formula for a class of anticipative transforms of Brownian motion which possesses exponential functionals as anticipating factors. Our result unifies existing formulas in earlier works. As an application, we also consider the law of Brownian motion perturbed by a positive weight of a fairly wide class, and prove its invariance under an anticipative transformation associated with the perturbation. In the course of our exploration, a disintegration formula for the Wiener measure related to exponential functionals plays a key role.