Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion

Abstract

The main result is a two-dimensional identity in law. Let (B t ,L t ) and ( t , t ) be two independent pairs of a linear Brownian motion with its local time at 0. Let A t = 0 t exp(2B s )ds. Then, for fixed t, the pair (sinh(B t ),sinh(L t )) has the same law as ((A t ),exp(-B t )(A t )), and also as (exp(-B t )(A t ),(A t )). This result is an extension of an identity in distribution due to Bougerol that concerned the first components of each pair. Some other related identities are also considered. AbstractWe present a two-dimensional extension of an identity in distribution due to Bougerol [Bou] that involves the exponential functional of a linear Brownian motion. Even though this identity does not extend at the level of processes, we point at further striking relations in this direction.

Bougerol’s identity in law and extensions

“…Bougerol's identity in law combined with the symmetry principle of André [And87, Gal08] yields the following identity in law (see e.g. [BeY12,BDY12a]): for every fixed l > 0,…”

“…A first 2-dimensional extension of Bougerol's identity was obtained by Bertoin, Dufresne and Yor in [BDY12a] (for a first draft, see also [DuY11]). With (L t , t ≥ 0) denoting the local time at 0 of B, we have:…”

“…With some elementary computations, from (1) (e.g. identifying the densities of both sides, for further details see [Vakth11,BDY12a]), we may obtain the Gauss-Laplace transform of the clock A t : for every x ∈ R, with a(x) ≡ arg sinh(x) ≡ log…”

Explicit Formulae in Probability and in Statistical Physics

“…With his prodigious knowledge of probability, he was able to use exponential functionals as a tool to re-derive or extend results obtained by other methods in other parts of the theory of stochastic processes. In the particular case where the Lévy process is a Brownian motion with drift, his work with H. Matsumoto led among other things to extensions of Bougerol's identity [10] and of Pitman's 2M − X theorem [2,6,34,35,38]. For more general Lévy processes, a dominant theme was the correspondence between exponential functionals and the semi-stable processes introduced by J. Lamperti [32].…”

A Link Between Bougerol’s Identity and a Formula Due to Donati-Martin, Matsumoto and Yor

“…Recently, there has been a renewed interest in generalizations of Bougerol's identity (1.1). Bertoin et al [3] presented a two-dimensional extension of (1.1) that involves some exponentional functional and the local time at 0 of a standard Wiener process. For another two-dimensional extension of (1.1), and even a three-dimensional one we refer to Vakeroudis [13,Sections 4.2 and 4.3].…”