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 $e^{B_{t}},\,t\ge 0$. 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.