Statistical inference for generalized Ornstein-Uhlenbeck processes

In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type $$X_{t} = e^{-\xi_{t}} \left( X_{0} + \int_{0}^{t} e^{\xi_{u-}} d u \right),$$ where $\xi_s$ is a L{\'e}vy process. Our primal goal is to estimate the characteristics of the L\'evy process $\xi$ from the low-frequency observations of the process $X$. We present a novel approach towards estimating the L{\'e}vy 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.