Time-changed Ornstein–Uhlenbeck process

The Ornstein–Uhlenbeck process is one of the most popular systems used for financial data description. However, this process has also been examined in the context of many other phenomena. In this paper we consider the so-called time-changed Ornstein–Uhlenbeck process, in which time is replaced by an inverse subordinator of general infinite divisible distribution. Time-changed processes nowadays play an important role in various fields of mathematical physics, chemistry, and biology as well as in finance. In this paper we examine the main characteristics of the time-changed Ornstein–Uhlenbeck process, such as the covariance function. Moreover, we also prove the formula for a generalized fractional Fokker–Planck equation that describes the one-dimensional probability density function of the analyzed system. For three cases of subordinators we show the special forms of obtained general formulas. Furthermore, we mention how to simulate the trajectory of the Ornstein–Uhlenbeck process delayed by a general inverse subordinator. (paper)