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

Let B = { B t } t ≥ 0 be a one-dimensional standard Brownian motion and denote by A t , t ≥ 0, the quadratic variation of e B t , t ≥ 0. The celebrated Bougerol’s identity in law (1983) asserts that, if β = { β t } t ≥ 0 is another Brownian motion independent of B , then β 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.