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

TL;DR

This paper introduces a method to estimate the driving Lévy process of multivariate CARMA models from discrete data, achieving asymptotic normality under specific sampling conditions.

Contribution

It develops a novel state space approach to recover the Lévy process from discrete observations, extending to approximate recovery and asymptotic analysis.

Findings

Exact recovery from continuous records

Asymptotic normality of estimators

Effective for decreasing sampling intervals

Abstract

We consider the parametric estimation of the driving L\'evy 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 L\'evy 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 L\'evy process from discrete-time observations of the MCARMA process. We show that, if the sampling interval $h=h_N$ is chosen dependent on $N$, the length of the observation horizon, such that $N h_N$ 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.