Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations
We consider the parametric estimation of the driving Lévy process of a multivariate continuous-time
autoregressive moving average (MCARMA) process, which is observed on the discrete time grid (0,h, 2h, ...).
Beginning with a new state space representation, we develop a method to recover the driving Lévy process
exactly from a continuous record of the observed MCARMA process. We use tools from numerical analysis
and the theory of infinitely divisible distributions to extend this result to allow for the approximate recovery
of unit increments of the driving Lévy process from discrete-time observations of the MCARMA process. We
show that, if the sampling interval h = hN is chosen dependent on N, the length of the observation horizon, such
that NhN converges to zero as N tends to infinity, then any suitable generalized method of moments estimator
based on this reconstructed sample of unit increments has the same asymptotic distribution as the one based on
the true increments, and is, in particular, asymptotically normally distributed.