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.

arXiv comment8 pages

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• Directed polymers and the quantum Toda lattice
• Exponential functionals of Brownian motion, II: Some related diffusion processes
• Interacting diffusions on positive definite matrices
• On some identities in law involving exponential functionals of Brownian motion and Cauchy variable