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

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 semimartingale $e^{B_t},t\ge 0$. The celebrated Bougerol’s identity in law (1983) asserts that, if $\mathit{\beta }={\{{\mathit{\beta }}_t\}}_{t\ge 0}$ is another Brownian motion independent of B, then ${\mathit{\beta }}_{A_t}$ 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 β 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.