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On some identities in law involving exponential functionals of Brownian motion and Cauchy random variable
در مورد برخی از هویت های قانونی مربوط به عملکردهای نمایی حرکت براونی و متغیر تصادفی کوشی-2020 Let B = {Bt }t≥0 be a one-dimensional standard Brownian motion, to which we associate the
exponential additive functional At = ∫ t
0 e2Bs ds, 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−Bt At }t≥0, the following identity in law holds:
(eBτ sinh x + β(Aτ ), CeBτ cosh x + βˆ(Aτ ), e−BτAτ ) (d)= ( sinh(x + Bτ ), C cosh(x + Bτ ), e−BτAτ ) ,
which extends an identity due to Bougerol (1983) in several aspects. Here β = {β(t)}t≥0 and
βˆ = {βˆ(t)}t≥0 are one-dimensional standard Brownian motions, C is a standard Cauchy random variable,
and B, β, βˆ 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 ( eBτ sinh x + 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 |
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