Inference for partially observed Riemannian Ornstein-Uhlenbeck diffusions of covariance matrices

TL;DR

This paper develops a Riemannian geometric framework for Ornstein-Uhlenbeck processes on covariance matrices and introduces a Bayesian inference method using MCMC with a novel diffusion bridge sampler, demonstrated on financial data.

Contribution

It generalizes Ornstein-Uhlenbeck processes to the Riemannian setting of covariance matrices and proposes a new MCMC-based Bayesian inference method accounting for geometric structure.

Findings

Successful implementation of the geometric inference method

Application to real financial data demonstrating effectiveness

Novel diffusion bridge sampler improves inference accuracy

Abstract

We construct a generalization of the Ornstein-Uhlenbeck processes on the cone of covariance matrices endowed with the Log-Euclidean and the Affine-Invariant metrics. Our development exploits the Riemannian geometric structure of symmetric positive definite matrices viewed as a differential manifold. We then provide Bayesian inference for discretely observed diffusion processes of covariance matrices based on an MCMC algorithm built with the help of a novel diffusion bridge sampler accounting for the geometric structure. Our proposed algorithm is illustrated with a real data financial application.