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

2017/01/26 by Stelzer, Robert, Vestweber, Johanna
#60G10 #60G51 #60J25 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1701.07859

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.

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