Maximum likelihood estimation for symmetric α-stable Ornstein–Uhlenbeck processes

Abstract

In this paper, we consider the maximum likelihood estimation for the symmetric [Formula: see text]-stable Ornstein–Uhlenbeck (S[Formula: see text]S-OU) processes based on discrete observations. Since the closed-form expression of maximum likelihood function is hard to obtain in the Lévy case, we choose a mixture of Cauchy and Gaussian distribution to approximate the probability density function (PDF) of the S[Formula: see text]S distribution. By means of transition function and Laplace transform, we construct an explicit approximate sequence of likelihood function, which converges to the likelihood function of S[Formula: see text]S distribution. Based on the approximation of likelihood function we give an algorithm for computing maximum likelihood estimation. We also numerically simulate some experiments which demonstrate the accuracy and stability of the proposed estimator.