Statistical inference for generalized Ornstein-Uhlenbeck processes

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

Беломестный Д. В., Панов В. А.

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.

Ключевые слова: Levy processes, generalized Ornstein-Uhlenbeck processes, exponential functional, Процессы Леви, Mellin transform, экспоненциальный функционал, преобразование Меллина, обобщённый процесс Орнштейна-Уленбека

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