On some identities in law involving exponential functionals of Brownian motion and Cauchy variable

Abstract

Let $B=\{B_t{\}}_{t\ge 0}$ be a one-dimensional standard Brownian motion, to which we associate the exponential additive functional $A_t={\int }_0^t{e}^{2B_s}ds,t\ge 0$. Starting from a simple observation of generalized inverse Gaussian distributions with particular sets of parameters, we show, with the help of a result by Matsumoto--Yor (2000), that for every $x\in \mathbb{R}$ and for every finite stopping time $\tau$ of the process $\{e^{-B_t}{A}_t{\}}_{t\ge 0}$, there holds the identity in law \begin{align*} \left( e^{B_τ}\!\sinh x+β(A_{τ}), \, Ce^{B_τ}\!\cosh x+\hatβ(A_{τ}), \, e^{-B_{τ}}\!A_{τ} \right) \stackrel{(d)}{=} \left( \sinh (x+B_{τ}), \, C\cosh (x+B_{τ}), \, e^{-B_{τ}}\!A_{τ} \right) , \end{align*} which extends an identity due to Bougerol (1983) in several aspects. Here $\beta =\{\beta (t){\}}_{t\ge 0}$ and $\hat{\beta }=\{\hat{\beta }(t){\}}_{t\ge 0}$ are one-dimensional standard Brownian motions, $C$ is a standard Cauchy variable, and $B$, $\beta$, $\hat{\beta }$ and $C$ are independent. Using an argument relevant to derivation of the above identity, we also present some invariance formulae for Cauchy variable involving an independent Rademacher variable.

43 pages. Changes from the first version are: positivity condition imposed on the stopping time $\tau$ is removed from Abstract, on which a remark is inserted in Remark 1.1; the assertion of Theorem 1.2 is fairly extended; two papers by Barndorff-Nielsen and two books are added for descriptions of GIG and related laws; a paper by Matsumoto--Yor (2003) is referred to in the newly added Remark A.1