Efficient maximum likelihood estimation for Lévy-driven Ornstein-Uhlenbeck processes

Abstract

We consider the problem of efficient estimation of the drift parameter of an Ornstein-Uhlenbeck type process driven by a Lévy process when high-frequency observations are given. The estimator is constructed from the time-continuous likelihood function that leads to an explicit maximum likelihood estimator and requires knowledge of the continuous martingale part. We use a thresholding technique to approximate the continuous part of the process. Under suitable conditions we prove asymptotic normality and efficiency in the Hájek-Le Cam sense for the resulting drift estimator. To obtain these results we prove an estimate for the Markov generator of a pure jump Lévy process. Finally, we investigate the finite sample behavior of the method and compare our approach to least squares estimation.