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Extensions of Bougerol’s identity in law and the associated anticipative path transformations

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  • Hariya, Yuu

Abstract

Let B={Bt}t≥0 be a one-dimensional standard Brownian motion and denote by At,t≥0, the quadratic variation of the geometric Brownian motion eBt,t≥0. Bougerol’s celebrated identity in law (1983) asserts that, if β={β(t)}t≥0 is another Brownian motion independent of B, then, for every fixed t>0, β(At) is identical in law with sinhBt. In this paper, we extend Bougerol’s identity to an identity in law for processes up to time t, which exhibits a certain invariance of the law of Brownian motion. The extension is described in terms of anticipative transforms of B involving At as an anticipating factor. A Girsanov-type formula for those transforms is shown. An extension of a variant of Bougerol’s identity is also presented.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:146:y:2022:i:c:p:311-334
    DOI: 10.1016/j.spa.2022.01.005
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Hariya, Yuu, 2024. "Invariance of Brownian motion associated with exponential functionals," Stochastic Processes and their Applications, Elsevier, vol. 167(C).

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