The stationary distribution of Ornstein–Uhlenbeck process with a two-state Markov switching

Abstract

ABSTRACTIn this article, we consider some characterizations for the stationary distribution of Ornstein–Uhlenbeck process with a two-state Markov switching. We show that if the drift coefficients α(1), α(2) are negative real numbers and diffusion coefficients σ(1), σ(2) are not equal to zero, then the stationary distributions of OU process with a two-state Markov switching have the density functions and the explicit Fourier transform of stationary density functions are obtained under some special cases. Furthermore, under some stronger assumptions for transition rate of the Markov chain, the Fourier transform of the density functions of stationary distribution can be approximated bywhere , πj is the invariant measure of Markov chain. Besides, some explicit expressions for the stationary distribution are presented and a numerical figure is used to illustrate our result.