Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type

In this article we propose a maximum likelihood methodology to estimate the parameters of a one-dimensional stationary process of Ornstein-Uhlenbeck type that is constructed via a self-decomposable distribution D. Our approach is based on the inversion of the characteristic function and the use of the classical or fractional discrete fast Fourier transform. The results are illustrated throughout an extensive simulation study. This includes the cases where D belongs to the gamma, tempered stable and normal inverse Gaussian family of distributions.