Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes

The class of multivariate Levy-driven autoregressive moving average (MCARMA) processes, the continuous-time analogs of the classical vector ARMA processes, is shown to be equivalent to the class of continuous-time state space models. The linear innovations of the weak ARMA process arising from sampling an MCARMA process at an equidistant grid are proved to be exponentially completely regular (mixing) under a mild continuity assumption on the driving Levy process. It is verified that this continuity assumption is satisfied in most practically relevant situations, including the case where the driving Levy process has a non-singular Gaussian component, is compound Poisson with an absolutely continuous jump size distribution or has an infinite Levy measure admitting a density around zero.