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Matsumoto-Yor processes on Jordan algebras

Reda Chhaibi, Manon Defosseux

TL;DR

The paper generalizes Matsumoto–Yor processes to Euclidean Jordan algebras, proving that a generalized geometric 2M-X Matsumoto–Yor process remains Markov on symmetric cones and satisfies a Dufresne-type identity. It develops a discrete Rider–Valkó–type block construction in Jordan-algebra settings, establishes intertwining-based proofs of the Matsumoto–Yor property and a discrete inverse-Wishart limit, and then passes to continuous time via a Stroock–Varadhan invariance principle on Lie groups, yielding continuous-time analogues. Special cases recover known matrix-valued results and yield new phenomena for the Lorentz cone and other symmetric cones, including potential octonionic extensions. The approach hinges on Jordan-algebra quadratic representations and intertwinings, unifying probabilistic and representation-theoretic viewpoints and offering a framework for perpetuity-type identities on symmetric cones with rich algebraic structure.

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.

Matsumoto-Yor processes on Jordan algebras

TL;DR

The paper generalizes Matsumoto–Yor processes to Euclidean Jordan algebras, proving that a generalized geometric 2M-X Matsumoto–Yor process remains Markov on symmetric cones and satisfies a Dufresne-type identity. It develops a discrete Rider–Valkó–type block construction in Jordan-algebra settings, establishes intertwining-based proofs of the Matsumoto–Yor property and a discrete inverse-Wishart limit, and then passes to continuous time via a Stroock–Varadhan invariance principle on Lie groups, yielding continuous-time analogues. Special cases recover known matrix-valued results and yield new phenomena for the Lorentz cone and other symmetric cones, including potential octonionic extensions. The approach hinges on Jordan-algebra quadratic representations and intertwinings, unifying probabilistic and representation-theoretic viewpoints and offering a framework for perpetuity-type identities on symmetric cones with rich algebraic structure.

Abstract

The process , where 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.

Paper Structure

This paper contains 29 sections, 30 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Background and literature
  3. Our objects of interest and contributions
  4. Intuition and statement of the main theorems
  5. Block setup for the discrete time Rider--Valkó's process
  6. Euclidean Jordan algebras, definitions
  7. Classification
  8. Main theorem and consequences in discrete setting
  9. Continuous time theorems
  10. Further comments and examples
  11. Considerations on Brownian motion on Lie groups
  12. Historical comments on Jordan algebras
  13. The example of the Lorentz cone
  14. Euclidean Jordan algebras
  15. Rank
  16. ...and 14 more sections

Key Result

Theorem 1.1

The process $\lambda = \left( \lambda_t \ ; \ t \geq 0 \right)$ defined for $t \geq 0$ by is a Markov process, whose natural filtration is strictly contained in that of $b$, i.e. for all $t > 0$, ${\mathcal{F}}_t^\lambda \varsubsetneq {\mathcal{F}}_t^b$. Furthermore, we have the conditional distribution

Theorems & Definitions (52)

  • Theorem 1.1: Pitman's Theorem pitman1975one
  • Theorem 1.2: Matsumoto--Yor matsumoto2000analogue
  • Theorem 1.3: Rider--Valkó rider2016matrix
  • proof : Pointers to proof
  • Definition 2.1
  • Theorem 2.2: Main Theorem, discrete time version
  • Definition 2.3
  • Theorem 2.4: Dufresne Identity for Jordan algebras, discrete version
  • Theorem 2.5: Main Theorem, continuous time version
  • Theorem 2.6: Dufresne Identity for Jordan algebras, continuous version
  • ...and 42 more