A multi-dimensional version of Lamperti’s relation and the Matsumoto–Yor processes

Thomas Gerard, Valentin Rapenne, Christophe Sabot and Xiaolin Zeng

Stochastic Processes and their Applications, 2024, vol. 175, issue C

Abstract: The distribution of a one-dimensional drifted Brownian motion conditioned on its first hitting time to 0 is the same as a three-dimensional Bessel bridge. By applying the time change in Lamperti’s relation to this result, Matsumoto and Yor (2001) showed a relation between Brownian motions with opposite drifts. In two subsequent papers (Matsumoto and Yor, 2000; 2001), they established a geometric lifting of the process 2M-B in Pitman’s theorem, known as the Matsumoto–Yor process. They also established an equality in law involving Inverse Gaussian distribution and its reciprocal (as processes), known as the Matsumoto–Yor property, by conditioning some exponential Wiener functional.

Date: 2024
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DOI: 10.1016/j.spa.2024.104401

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