Stationary density function for a random evolution driven by a Markov-switching Ornstein–Uhlenbeck process with finite velocity

Abstract

Abstract In this paper, we consider a new telegraph process of Ornstein–Uhlenbeck type. The process is obtained by generalizing the telegraph process in a similar manner to how the Ornstein–Uhlenbeck process was obtained from the Wiener process, namely by adding a drift coefficient proportional to a displacement from the origin. This process was first introduced by Ratanov in [N. Ratanov, Ornstein–Uhlenbeck process of bounded variation, Methodol. Comput. Appl. Probab. 23 2021, 925–946]. We obtain the infinitesimal operator of this process and we present formulas for finding its stationary probability density. We consider both the symmetric and asymmetric cases.