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

Yuu Hariya

Stochastic Processes and their Applications, 2020, vol. 130, issue 10, 5999-6037

Abstract: 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.

Keywords: Brownian motion; Exponential functional; Bougerol’s identity; Cauchy random variable; Generalized inverse Gaussian distribution (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2020.05.001

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