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 to 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 provide a Dufresne-type identity for a perpetuity. In particular, the paper provides a generalization to any symmetric cone of the initial matrix generalization of the Matsumoto-Yor process and Dufresne identity by Rider-Valk\'o.