Wishart Processes

TL;DR

This paper reviews Wishart processes, a class of matrix-valued stochastic processes with applications in finance, discussing their mathematical properties, solutions to their defining equations, and their use in modeling covariance matrices.

Contribution

It provides a comprehensive overview of Wishart processes, including existence, uniqueness, and solution representations, highlighting their applications in financial modeling.

Findings

Existence and uniqueness of strong solutions discussed

Some solutions expressed as squares of Ornstein-Uhlenbeck processes

Applications in modeling interest rates and covariance matrices

Abstract

Based on a student research project this article gives a short review on Wishart processes. A Wishart procces is a matrix valued continuous time stochastic process with a marginal Wishart distribution. The Wishart distribution is a matrix variate generalization of the chi-squared distribution. Since Wishart processes are defined as a solution to a stochastic differential equation, the existence and uniqueness of strong solutions will be discussed comprehensively. It is also shown that some solutions of the stochastic differential equation can be expressed as squares of matrix variate Ornstein-Uhlenbeck processes. Wishart processes have the property of being symmetric positive definite and are therefore heavily used for modeling interest rates or the covariance matrix in stochastic volatility models.