YOR: Matsumoto–Yor Processes on Jordan Algebras

• YOR is a generalization of classical Matsumoto–Yor processes defined on Euclidean Jordan algebras, extending exponential Brownian functionals to symmetric cones.
• It establishes novel Markovian structures via left-invariant diffusions and intertwining conditional laws described by generalized inverse Gaussian distributions on cones.
• The framework bridges stochastic calculus, random matrix theory, and perpetuity identities, offering actionable insights for applications in finance and integrable systems.

YOR refers, in the context of stochastic processes and their generalizations, to the class of Matsumoto–Yor processes on Jordan algebras, and more broadly to associated Markovian structures and Dufresne-type identities extended to symmetric cones and multivariate frameworks (Chhaibi et al., 2024). The subject builds on classical exponential functionals of Brownian motion—central in probabilistic analysis, integrable systems, financial mathematics, and random matrix theory—and generalizes them from the real line to the rich setting of Euclidean Jordan algebras and their symmetric cones.

1. Euclidean Jordan Algebras and Symmetric Cones

A Euclidean Jordan algebra is a finite-dimensional real commutative algebra $V$ equipped with a bilinear product $(x, y) \mapsto x \cdot y$ satisfying the Jordan identity $x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$, a distinguished unit $e$, and a compatible inner product. Every $x \in V$ admits a spectral decomposition $x = \sum_{i=1}^r \lambda_i c_i$, where $\{c_i\}$ are orthogonal idempotents (a Jordan frame), and $r$ is the rank.

The symmetric cone $\Omega = V_+ \subset V$ consists of elements with strictly positive eigenvalues. Such cones are open, self-dual, and homogeneous under the automorphism group $G = \mathrm{Aut}(\Omega)$. Jordan algebra tools include the quadratic representation $(x, y) \mapsto x \cdot y$0, where $(x, y) \mapsto x \cdot y$1, and the cone’s exponential map, which in spectral coordinates is computed entrywise. These structures enable the definition of diffusion processes and functionals generalizing classical Brownian motion (Chhaibi et al., 2024).

2. The Classical Matsumoto–Yor Process and Its Markov Property

The canonical one-dimensional Matsumoto–Yor process arises from a real Brownian motion $(x, y) \mapsto x \cdot y$2 with drift $(x, y) \mapsto x \cdot y$3, via the exponential functional $(x, y) \mapsto x \cdot y$4 and the process $(x, y) \mapsto x \cdot y$5. These processes are Markovian and possess filtrations strictly smaller than that of the driving Brownian motion.

Crucially, the conditional law $(x, y) \mapsto x \cdot y$6 is that of a generalized inverse Gaussian (GIG) distribution, and the process $(x, y) \mapsto x \cdot y$7 solves

$(x, y) \mapsto x \cdot y$8

This has far-reaching implications, linking path decompositions, special function theory, and representation theory (Chhaibi et al., 2024).

3. Matsumoto–Yor Processes on Jordan Algebras: Construction and Structure

The generalization to Jordan algebras proceeds via a left-invariant diffusion $(x, y) \mapsto x \cdot y$9 on the automorphism group $x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$0, defined by the Stratonovich SDE

$x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$1

with $x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$2 spanning the Jordan algebra, $x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$3 independent BMs, and $x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$4 a parameter.

One builds three interlinked processes:

• The integral process $x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$5,
• The “AX+B” process $x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$6,
• The Matsumoto–Yor process $x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$7.

In exponential coordinates ($x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$8), the process has the explicit form

$x \cdot (x^2 \cdot y) = (x^2) \cdot (x \cdot y)$9

generalizing the real case via spectral calculus in the Jordan algebra (Chhaibi et al., 2024).

4. Markovianity, Conditional Laws, and Infinitesimal Generator

The Markov property persists in the Jordan algebra context: $e$0 is Markov on the cone, with filtration strictly smaller than that of the driving $e$1. Conditional on the past trajectory $e$2, the process $e$3 is governed by a multivariate GIG law $e$4.

The infinitesimal generator acting on smooth compactly supported $e$5 is

$e$6

where $e$7 is the Jordan-gradient, and the first term applies the trace to the Hessian weighted by $e$8. This reduces to the classical generator in the case $e$9 (Chhaibi et al., 2024).

5. Discrete-Time AX+B Models and Dufresne-Type Identities

A discrete-time analog involves i.i.d. increments $x \in V$0 in $x \in V$1 with law $x \in V$2, and two coupled Markov chains:

1. $x \in V$3, $x \in V$4,
2. $x \in V$5, $x \in V$6.

Here, $x \in V$7 is Markov, and the conditional law of $x \in V$8, given $x \in V$9, is again a multivariate GIG. This leads to a Dufresne-type identity: the perpetuity

$x = \sum_{i=1}^r \lambda_i c_i$0

is inverse-Wishart distributed on $x = \sum_{i=1}^r \lambda_i c_i$1, under minimal moment conditions (Chhaibi et al., 2024).

6. Rank-One and Higher-Rank/Matrix Specializations

Special cases include:

• Rank One ($x = \sum_{i=1}^r \lambda_i c_i$2): Recovers the classical Matsumoto–Yor process $x = \sum_{i=1}^r \lambda_i c_i$3, with the Dufresne law $x = \sum_{i=1}^r \lambda_i c_i$4.
• Matrix Case ($x = \sum_{i=1}^r \lambda_i c_i$5): Yields the Rider–Valkó matrix Matsumoto–Yor process $x = \sum_{i=1}^r \lambda_i c_i$6 and the matrix Dufresne identity ($x = \sum_{i=1}^r \lambda_i c_i$7 is inverse-Wishart).

The framework extends verbatim to other symmetric cones associated to Lorentz, quaternionic, and octonionic Jordan algebras, with all Markov and intertwining properties preserved (Chhaibi et al., 2024).

7. Structural and Unifying Principles

The Jordan-algebraic Matsumoto–Yor process unifies and extends the stochastic calculus, spectral theory, and Markov intertwinings known in dimension one to all symmetric cones. The continuous processes appear as scaling limits of discrete-time “AX+B” chains, with all conditional, marginal, and intertwining properties preserved. These findings generalize, for example, the geometric interpretations linked to infinite-dimensional limits on symmetric spaces (Bougerol, 2014) and have notable analogs in the matrix Dufresne law, perpetuity theory, and noncommutative harmonic analysis.

The extension introduces new structures and functional identities that interpolate between classical exponential functionals, random matrix theory, and the rich geometry of symmetric cones. The Markov property, conditional law description via GIG distributions on the cone, and generalized Dufresne identities collectively form a new cornerstone in the probabilistic analysis on Jordan algebraic structures.

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Key Reference:

“Matsumoto-Yor processes on Jordan algebras” (Chhaibi et al., 2024)