On two-dimensional extensions of Bougerol's identity in law

TL;DR

This paper generalizes Bougerol's identity in law to two-dimensional cases with non-zero local times, providing a simpler proof and broader understanding of this probabilistic identity.

Contribution

It extends the two-dimensional Bougerol's identity to arbitrary levels of local times, broadening the scope of the original identity and offering a new, elementary proof.

Findings

Generalization of Bougerol's identity to non-zero local times

Simplified proof of the original two-dimensional extension

Enhanced understanding of the identity's underlying structure

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 $e^{B_{t}},\,t\ge 0$. The celebrated Bougerol's identity in law (1983) asserts that, if $\beta =\{ \beta _{t}\} _{t\ge 0}$ is another Brownian motion independent of $B$, then $\beta _{A_{t}}$ has the same law as $\sinh B_{t}$ for every fixed $t>0$. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving as the second coordinates the local times of $B$ and $\beta $ at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.