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We consider the parametric estimation of the driving L\'evy process of amultivariate continuous-time autoregressive moving average (MCARMA) process,which is observed on the discrete time grid $(0,h,2h,...)$. Beginning with anew state space representation, we develop a method to recover the drivingL\'evy process exactly from a continuous record of the observed MCARMA process.We use tools from numerical analysis and the theory of infinitely divisibledistributions to extend this result to allow for the approximate recovery ofunit increments of the driving L\'evy process from discrete-time observationsof the MCARMA process. We show that, if the sampling interval $h=h_N$ is chosendependent on $N$, the length of the observation horizon, such that $N h_N$converges to zero as $N$ tends to infinity, then any suitable generalizedmethod of moments estimator based on this reconstructed sample of unitincrements has the same asymptotic distribution as the one based on the trueincrements, and is, in particular, asymptotically normally distributed.

Parametric estimation of the driving L\'evy process of multivariate CARMA processes from discrete observations