The multivariate ARMA/CARMA transformation relation

Abstract: A transformation relation between multivariate ARMA and CARMA processes is derived through a discretization procedure. This gives a direct relationship between the discrete time and continuous time analogues, serving as the basis for an estimation method for multivariate CARMA models. We will see that the autoregressive coefficients, making up the deterministic part of a multivariate CARMA model, are entirely given by the transformation relation. An Euler discretization convergence rate of jump diffusions is found for the case of small jumps of infinite variation. This substantiates applying the transformation relation for estimation of multivariate CARMA models driven by NIG-L\'evy processes. A two-dimensional CAR model is fit to stratospheric temperature and wind data, as an example of how to apply the transformation relation in estimation methods.