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

ABSTRACT

In 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 by where , π_j 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.

KEYWORDS:

• Fourier transform
• Markov chain
• Ornstein–Uhlenbeck process
• Stationary density

MATHEMATICS SUBJECT CLASSIFICATION:

• 60H10
• 37A25

Acknowledgments

The authors are grateful to Prof. Zhen-Qing Chen, Dr. Guohuang Zhao, and anonymous referees for their valuable comments and suggestions which led to improvements in this article. Research of the first author was partially supported by National Natural Science Foundation of China (Nos. 11201062, 11401093, 11471071, 11571071), the Fundamental Research Funds for the Central Universities.