Exact long time behavior of some regime switching stochastic processes

Regime switching processes have proved to be indispensable in the modeling of various phenomena, allowing model parameters that traditionally were considered to be constant to fluctuate in a Markovian manner in line with empirical findings. We study diffusion processes of Ornstein–Uhlenbeck type where the drift and diffusion coefficients a and b are functions of a Markov process with a stationary distribution π on acountable state space. Exact long time behavior is determined for the three regimes corresponding to the expected drift: Eπa (·)>0,=0,<0, respectively. Alongside we provide exact time limit results for integrals of form ∫t0b2(Xs)e−2∫tsa(Xr)drds for the three different regimes. Finally, we demonstrate natural applications of the findings in terms of Cox–Ingersoll–Ross diffusion and deterministic SIS epidemic models inMarkovian environments. The time asymptotic behaviors are naturally expressed in terms of solutions tothe well-studied fixed-point equation in law Xd=AX+B with X⊥⊥(A, B) .