Explicit representation of characteristic function of tempered α‐stable Ornstein–Uhlenbeck process

Abstract

The Ornstein–Uhlenbeck process is a fundamental tool in various applications ranging from finance where it is known under Vasiček process name to fractional diffusion in the context, for instance, Klein–Kramers dynamics. In this paper, we define the Ornstein–Uhlenbeck process driven by a symmetric tempered α‐stable process. We investigate its properties in the non‐stationary case and its long time limit. Moreover, we provide an explicit representation of its characteristic function in terms of special functions, that is, generalized hypergeometric function and its special case hypergeometric Gauss function. Thus, using probabilistic arguments, we provide a new identity between those two functions. Our calculations are also confirmed by numerical simulations of this new process. We discuss the Fokker–Planck type equation describing the probability density function of our model as a solution of the fractional differential equation with the so‐called tempered fractional derivative.