Matsumoto-Yor processes on Jordan algebras

Abstract

The process $({\int }_0^t{e}^{2b_s-b_t}ds;t\ge 0)$, where $b$ is a real Brownian motion, is known as the geometric 2M-X Matsumoto--Yor process. Remarkably, it enjoys the Markov property. We provide a generalization of this process in the context of Jordan algebras, and we prove the Markov property for this generalization. Our Markov process occurs as a limit of discrete-time AX+B Markov chains on the cone of squares whose invariant probability measures classically yield a Dufresne-type identity for a perpetuity. In particular, the paper provides a generalization to any symmetric cone of the matrix--valued generalization of the Matsumoto--Yor process and Dufresne identity initially developed by Rider--Valkó.

arXiv commentv1: Preliminary version. v2: Polished version. v3: Submitted version, 32 pages

References in corpus (5)

• Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
• Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices
• Quantum $S{L}_2$, Infinite curvature and Pitman's 2M-X theorem