Picard iterations for diffusions on symmetric matrices

TL;DR

This paper extends the Wishart process by analyzing matrix-valued diffusions using Picard iterations, establishing existence and uniqueness of solutions in the space of symmetric matrices.

Contribution

It introduces a novel Picard iteration approach tailored for matrix diffusions, highlighting the natural emergence of Lipschitz conditions in this setting.

Findings

Proves existence and uniqueness of matrix diffusion solutions

Develops Picard iteration method for symmetric matrix processes

Clarifies the role of operator properties in matrix stochastic differential equations

Abstract

Matrix-valued stochastic processes have been of significant importance in areas such as physics, engineering and mathematical finance. One of the first models studied has been the so-called Wishart process, which is described as the solution of a stochastic differential equation in the space of matrices. In this paper we analyze natural extensions of this model, and prove the existence and uniqueness of the solution. We do this by carrying out a Picard iteration technique in the space of symmetric matrices. This approach takes into account the operator character of the matrices, which helps to corroborate how the Lipchitz conditions also arise naturally in this context.