Statistical inference for generalized Ornstein-Uhlenbeck processes

Abstract

In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type \[X_{t}=e^{-\xi_{t}}\left(X_{0}+\int_{0}^{t}e^{\xi_{u-}}du \right), \] where $\xi_{s}$ is a Lévy process. Our primal goal is to estimate the characteristics of the Lévy process $\xi$ from the low-frequency observations of the process $X$. We present a novel approach towards estimating the Lévy triplet of $\xi$, which is based on the Mellin transform technique. It is shown that the resulting estimates attain optimal minimax convergence rates. The suggested algorithms are illustrated by numerical simulations.