The Stationary Distributions of Doubly Skew Ornstein-Uhlenbeck Processes and Markov-modulated Skew Ornstein-Uhlenbeck Processes

In this paper, we consider the stationary density function of the doubly skew Ornstein-Uhlenbeck process. We present the explicit formula for the stationary density function and show that this process is positive Harris recurrent and geometrically ergodic. We expand our method to the more general cases in which the multiple parameters are present and we try to consider the stability of the skew Ornstein-Uhlenbeck process whose parameters depend on a finite-state and irreducible continuous-time Markov chain. Then we offer the stationary distribution equation of this bivariate process through their infinitesimal generator and the explicit stationary distributions are list as a special case.