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

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  • Hariya, Yuu

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.

Suggested Citation

  • Hariya, Yuu, 2020. "On some identities in law involving exponential functionals of Brownian motion and Cauchy random variable," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 5999-6037.
  • Handle: RePEc:eee:spapps:v:130:y:2020:i:10:p:5999-6037
    DOI: 10.1016/j.spa.2020.05.001
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    References listed on IDEAS

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    1. Barndorff-Nielsen, O. & Blæsild, P. & Halgreen, C., 1978. "First hitting time models for the generalized inverse Gaussian distribution," Stochastic Processes and their Applications, Elsevier, vol. 7(1), pages 49-54, March.
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    Cited by:

    1. Yuu Hariya, 2022. "Integral Representations for the Hartman–Watson Density," Journal of Theoretical Probability, Springer, vol. 35(1), pages 209-230, March.
    2. Hariya, Yuu, 2022. "Extensions of Bougerol’s identity in law and the associated anticipative path transformations," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 311-334.

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