Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes

Abstract

We consider nonparametric estimation of the Lévy measure of a hidden Lévy process driving a stationary Ornstein-Uhlenbeck process which is observed at discrete time points. This Lévy measure can be expressed in terms of the canonical function of the stationary distribution of the Ornstein-Uhlenbeck process, which is known to be self-decomposable. We propose an estimator for this canonical function based on a preliminary estimator of the characteristic function of the stationary distribution. We provide a support-reduction algorithm for the numerical computation of the estimator, and show that the estimator is asymptotically consistent under various sampling schemes. We also define a simple consistent estimator of the intensity parameter of the process. Along the way, a nonparametric procedure for estimating a self-decomposable density function is constructed, and it is shown that the Ornstein-Uhlenbeck process is β-mixing. Some general results on uniform convergence of random characteristic functions are included.