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

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.

Sabot and Zeng (2020) generalized some results on drifted Brownian motion conditioned on its first hitting. More precisely, they introduced a family of Brownian semimartingales with interacting drifts, for which when conditioned on the vector $\tau$ (the hitting times to 0 of each component), their joint law is the same as for independent three-dimensional Bessel bridges. The distribution of $\tau$ is a generalization of Inverse Gaussian distribution in multi-dimension and it is related to a random potential $\beta$ that appears in the study of the Vertex Reinforced Jump Process.

The aim of this paper is to generalize the results of Matsumoto and Yor (2001, 2000) in the context of these interacting Brownian semimartingales. We apply a Lamperti-type time change to the previous family of interacting Brownian motions and we obtain a multi-dimensional opposite drift theorem. Moreover, we also give a multi-dimensional counterpart of the Matsumoto–Yor process and its intertwining relation with interacting geometric Brownian motions.