Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices

TL;DR

This paper extends classical stochastic theorems to positive definite matrices, establishing new identities and recursions for matrix-valued random walks with Beta type II distributed steps.

Contribution

It introduces matrix analogues of the Matsumoto-Yor and Dufresne theorems for multiplicative random walks on positive definite matrices.

Findings

Established matrix versions of the Matsumoto-Yor and Dufresne theorems.

Derived explicit solutions for matrix stochastic recursions.

Connected matrix identities to Beta type II distributions.

Abstract

We establish analogues of the geometric Pitman $2M-X$ theorem of Matsumoto and Yor and of the classical Dufresne identity, for a multiplicative random walk on positive definite matrices with Beta type II distributed increments. The Dufresne type identity provides another example of a stochastic matrix recursion, as considered by Chamayou and Letac (J. Theoret. Probab. 12, 1999), that admits an explicit solution.