Multivariate COGARCH(1,1) processes

TL;DR

This paper introduces multivariate COGARCH(1,1) processes as continuous-time models for multidimensional heteroskedastic data, analyzing their probabilistic properties and conditions for stationarity and moment finiteness.

Contribution

It provides a novel multivariate COGARCH(1,1) model driven by a single Lévy process with explicit conditions for stationarity and moments, extending univariate models to higher dimensions.

Findings

Established conditions for the existence of a stationary distribution.

Derived explicit formulas for first and second moments.

Analyzed stationarity and second-order structure of increments.

Abstract

Multivariate $\operatorname {COGARCH}(1,1)$ processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate L\'{e}vy process and the latent time-varying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices. After defining the $\operatorname {COGARCH}(1,1)$ process, we analyze its probabilistic properties. We show a sufficient condition for the existence of a stationary distribution for the stochastic covariance matrix process and present criteria ensuring the finiteness of moments. Under certain natural assumptions on the moments of the driving L\'{e}vy process, explicit expressions for the first and second-order moments and (asymptotic) second-order stationarity of the covariance matrix process are obtained. Furthermore, we study the stationarity and second-order structure of the increments of the multivariate $\operatorname {COGARCH}(1,1)$ process and their "squares".