Abstract
Let be a one-dimensional standard Brownian motion and denote by , the quadratic variation of semimartingale . The celebrated Bougerol’s identity in law (1983) asserts that, if is another Brownian motion independent of B, then has the same law as for every fixed . Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving, as the second coordinates, the local times of B and β at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.
Funding Statement
The research of Y. Hariya was supported in part by JSPS KAKENHI Grant Number 22K03330.
Acknowledgments
The authors wish to thank the anonymous referees for their constructive comments, especially on the literature concerning Dufresne’s identity in law as cited at the end of Section 1.
Citation
Yuu Hariya. Yohei Matsumura. "On two-dimensional extensions of Bougerol’s identity in law." Electron. Commun. Probab. 28 1 - 7, 2023. https://doi.org/10.1214/23-ECP510
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