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

Let B={Bt}t≥0 be a one-dimensional standard Brownian motion, to which we associate the exponential additive functional At=∫0te2Bsds,t≥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 and Yor (2000), that, for every x∈R and for every positive and finite stopping time τ of the process {e−BtAt}t≥0, the following identity in law holds: eBτsinhx+β(Aτ),CeBτcoshx+β̂(Aτ),e−BτAτ=(d)sinh(x+Bτ),Ccosh(x+Bτ),e−BτAτ, which extends an identity due to Bougerol (1983) in several aspects. Here β={β(t)}t≥0 and β̂={β̂(t)}t≥0 are one-dimensional standard Brownian motions, C is a standard Cauchy random variable, and B, β, β̂ and C are independent. The derivation of the above identity provides another proof of Bougerol’s identity in law; moreover, a similar reasoning also enables us to obtain another extension for the three-dimensional random variable eBτsinhx+β(Aτ),eBτ,Aτ. By using an argument relevant to the derivation of those results, some invariance formulae for the Cauchy random variable C involving an independent Rademacher random variable, are presented as well.