# Geometric Ergodicity of the multivariate COGARCH(1,1) Process

## TL;DR

This paper establishes conditions for the existence, uniqueness, and geometric ergodicity of the stationary distribution of the multivariate COGARCH(1,1) process, using Foster-Lyapunov methods and geometric considerations.

## Contribution

It provides new sufficient conditions for ergodicity and moments of the multivariate COGARCH(1,1) process, especially under compound Poisson driving Lévy processes.

## Key findings

- Conditions for existence of a unique stationary distribution.
- Criteria for geometric ergodicity of the process.
- Applicability to processes driven by compound Poisson Lévy processes.

## Abstract

For the multivariate COGARCH(1,1) volatility process we show sufficient conditions for the existence of a unique stationary distribution, for the geometric ergodicity and for the finiteness of moments of the stationary distribution by a Foster-Lyapunov drift condition approach. The test functions used are naturally related to the geometry of the cone of positive semi-definite matrices and the drift condition is shown to be satisfied if the drift term of the defining stochastic differential equation is sufficiently `negative'. We show easily applicable sufficient conditions for the needed irreducibility and aperiodicity of the volatility process living in the cone of positive semidefinite matrices, if the driving L\'evy process is a compound Poisson process.