A Mean-Reverting SDE on Correlation matrices

TL;DR

This paper introduces a novel mean-reverting stochastic differential equation on correlation matrices, extending Wright-Fisher diffusion, with applications in financial asset dependence modeling.

Contribution

It develops a new SDE on correlation matrices, analyzes its uniqueness and ergodic properties, and proposes high-order simulation schemes for financial modeling.

Findings

Established conditions for SDE uniqueness

Described ergodic behavior of the process

Developed second-order weak convergence discretization schemes

Abstract

We introduce a mean-reverting SDE whose solution is naturally defined on the space of correlation matrices. This SDE can be seen as an extension of the well-known Wright-Fisher diffusion. We provide conditions that ensure weak and strong uniqueness of the SDE, and describe its ergodic limit. We also shed light on a useful connection with Wishart processes that makes understand how we get the full SDE. Then, we focus on the simulation of this diffusion and present discretization schemes that achieve a second-order weak convergence. Last, we explain how these correlation processes could be used to model the dependence between financial assets.