Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion

TL;DR

This paper extends Bougerol's identity to a two-dimensional setting involving the exponential functional of linear Brownian motion, revealing new distributional relations despite not extending to process level.

Contribution

It introduces a novel two-dimensional extension of Bougerol's identity for exponential functionals of Brownian motion, highlighting new distributional relationships.

Findings

Established a two-dimensional distributional extension of Bougerol's identity.

Identified additional notable relations related to the original identity.

Clarified limitations in extending the identity to process level.

Abstract

We present a two-dimensional extension of an identity in distribution due to Bougerol \cite{Bou} that involves the exponential functional of a linear Brownian motion. Even though this identity does not extend at the level of processes, we point at further striking relations in this direction.