Multivariate Lévy-driven continuous time autoregressive moving average processes are the continuous time analogues of the well-known ARMA processes. Formally, they are defined as stationary solutions to (possibly high order) linear differential equations driven by a Lévy process. In this talk we will first discuss in detail the proper definition and the equivalence to general order continuous-time state space models. Then we will summarise important probabilistic properties which are related to the statistical inference.

Finally, we will establish that under natural assumptions a quasi-maximum likelihood approach can be used to estimate the autoregressive and moving average parameters of a multivariate CARMA process based on discrete time equidistant observations. We will illustrate the estimation scheme by a simulation study and a data example.