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 $X_{t} = e^{-ξ_{t}} \left( X_{0} + \int_{0}^{t} e^{ξ_{u-}} d u \right),$ where $ξ_s$ is a L{é}vy process. Our primal goal is to estimate the characteristics of the Lévy process $ξ$ from the low-frequency observations of the process $X$. We present a novel approach towards estimating the L{é}vy triplet of $ξ,$ 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.

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