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

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 Lévy process is a compound Poisson process.

Moment‐based estimation for the multivariate COGARCH(1,1) process

“…Sufficient conditions for the existence of a unique stationary distribution of (Y t ) t∈R + , geometric ergodicity and for the finiteness of moments of order p of the stationary distribution have recently been given in Stelzer and Vestweber (2019). We state these conditions in the next theorem, which are conditions (i), (iv), and (v) of theorem 4.3 in Stelzer and Vestweber (2019).…”

“…In the univariate case, Fasen (2010) proved geometric ergodicity results for the COGARCH process (in fact, their results apply to a wider class of Lévy driven models). Recently, Stelzer and Vestweber (2019) derived sufficient conditions for the existence of a unique stationary distribution, for the geometric ergodicity, and for the finiteness of moments of the stationary distribution in the MUCOGARCH process. These results imply ergodicity and strong mixing of the log-price process (G i ) ∞ i=1 , thus paving the way for statistical inference.…”