Fractional lower order covariance based-estimator for Ornstein-Uhlenbeck process with stable distribution

Piotr Kruczek ; Wojciech Żuławiński ; Patrycja Pagacz ; Agnieszka Wyłomańska

Mathematica Applicanda, Tome 47 (2019) no. 2, pp. 259-292.

Résumé

The Ornstein-Uhlenbeck model is one of the most popular stochastic processes. It has found many interesting applications including physical phenomena. However, for many real data, the classical Ornstein-Uhlenbeck process cannot be applied. It is related to the fact that for many phenomena the vectors of observations exhibit so-called heavy-tailed behaviour. In such cases, the modifications of the classical models need to be used. In this paper, we analyze the Ornstein-Uhlenbeck process based on stable distribution. This distribution is one of the most classical members of the heavy-tailed class of distributions. In the literature, one can find various applications of stable processes. However, the heavy-tailed property implies that the classical methods of estimation and statistical investigation cannot be applied. In this paper, we propose a new method of estimation of stable Ornstein-Uhlenbeck process. This technique is based on the alternative measure of dependence, called fractional lower order covariance, which replaces the classical covariance for infinite-variance distribution. The proposed research is a continuation of the authors' previous studies, where the measure called covariation was proposed as the base for the estimation technique. We introduce the stable Ornstein-Uhlenbeck process and remind its main properties. In the main part, we define the new estimator of the of the parameters for discrete representation of Ornstein-Uhlenbeck process. Its effectiveness is checked by Monte Carlo simulations.

Détail

DOI : 10.14708/ma.v47i2.6506