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{\tau}}!\sinh x+\beta (A_{\tau }), \, Ce^{B_{\tau}}!\cosh x+\hat{\beta}(A_{\tau }), \, e^{-B_{\tau }}!A_{\tau } \right) \stackrel{(d)}{=} \left( \sinh (x+B_{\tau }), \, C\cosh (x+B_{\tau }), \, e^{-B_{\tau }}!A_{\tau } \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.