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 semi-definite matrices, if the driving Lévy process is a compound Poisson process.

摘要

对于多元 COGARCH(1,1) 波动率过程，我们通过 Foster-Lyapunov 漂移条件方法展示了存在唯一平稳分布、几何遍历性和平稳分布矩的有限性的充分条件。使用的测试函数自然与正半定矩阵的锥体的几何形状相关，如果定义的随机微分方程的漂移项足够“负”，则表明满足漂移条件。如果驱动的 Lévy 过程是复合 Poisson 过程，我们展示了存在于半正定矩阵锥中的波动过程所需的不可约性和非周期性的容易适用的充分条件。