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Statistical inference for generalized Ornstein-Uhlenbeck processes

Electronic journal of statistics. 2015. Vol. 9. No. 2. P. 1974-2006.

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.