Geometric ergodicity of the multivariate COGARCH(1,1) process

Geometric ergodicity of affine processes on cones

“…• If K is the cone of d × d symmetric positive definite matrices, one can also build the jumps by taking the outer products of jumps in R d and the above results remain true provided that they have a strictly positive density on R d (in a neighbourhood of zero). This can be shown by taking not only one jump after time T as in the above proof but at least d jumps and arguing similar to [81,Theorem 4.6].…”

“…So Q = 0 in addition to the assumptions made so far. The following sufficient conditions for irreducibility and aperiodicity are inspired by similar results of [81] for a non-linear SDE in the cone of positive semi-definite matrices. Proposition 4.5.…”

“…• Similarly to [81,Corollary 4.7] one can relax the conditions to demanding that m or η, µ(•) for one η ∈ K • have an absolutely continuous component with a strictly positive density only in a neighbourhood of zero in K • . Then X is irreducible with respect to the Lebesgue measure restricted to a neighbourhood of B −1 b and aperiodic.…”

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

“…Sufficient conditions for the existence of a unique stationary distribution of , 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 , thus paving the way for statistical inference.…”