Ornstein-Uhlenbeck Process Delayed by Gamma Subordinator

Abstract

The Ornstein-Uhlenbeck (OU) process is one of the most popular stochastic system applied in many different fields of studies. It was introduced in 1930 and can be considered as a continuous extension of the autoregressive model of order one, AR(1). Furthermore, the OU process in finance it is known as the Vasicek model and is mainly used in interest rate modelling. Furthermore, it is deeply studied and its main properties are well known. However, many real data exhibit some properties of the OU process although they cannot be directly modelled with this classical system. This is in case when certain characteristics adequate to the OU process are visible in the data however other properties of the classical model change. In such case the subordination scenario can be considered. In general, the subordination it is a time change of the original process. In this paper we consider the Ornstein-Uhlenbeck process delayed (subordinated) by Gamma subordinator. The Gamma subordinator is Levy process of Gamma distribution. The main properties are studied, like the influence of the initial condition on the stationarity of the new subordinated process. Moreover, the formulas for the expected value and the autocovariance are derived. Furthermore, the simulation procedures and estimation algorithms are proposed.