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

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 $τ$ 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 $β=\{ β(t)\} _{t\ge 0}$ and $\hatβ=\{ \hatβ(t)\} _{t\ge 0}$ are one-dimensional standard Brownian motions, $C$ is a standard Cauchy variable, and $B$, $β$, $\hatβ$ 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.

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