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A Girsanov-Type Formula for a Class of Anticipative Transforms of Brownian Motion Associated with Exponential Functionals

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  • Yuu Hariya

    (Tohoku University)

Abstract

In this paper, with the help of a result by Matsumoto and Yor (Nagoya Math J 159:125–166, 2000), we prove a Girsanov-type formula for a class of anticipative transforms of Brownian motion which possesses exponential functionals as anticipating factors. Our result unifies existing formulas in earlier works. As an application, we also consider the law of Brownian motion perturbed by a positive weight of a fairly wide class and prove its invariance under an anticipative transformation associated with the perturbation. In the course of our exploration, a disintegration formula for the Wiener measure related to exponential functionals plays a key role.

Suggested Citation

  • Yuu Hariya, 2025. "A Girsanov-Type Formula for a Class of Anticipative Transforms of Brownian Motion Associated with Exponential Functionals," Journal of Theoretical Probability, Springer, vol. 38(1), pages 1-23, March.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:1:d:10.1007_s10959-024-01383-9
    DOI: 10.1007/s10959-024-01383-9
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    References listed on IDEAS

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