## Exact long time behavior of some regime switching stochastic processes.(English)Zbl 1455.60106

Summary: 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 $$\pi$$ on a countable state space. Exact long time behavior is determined for the three regimes corresponding to the expected drift: $$E_{\pi}a(\cdot )>0, =0,<0$$, respectively. Alongside we provide exact time limit results for integrals of form $$\int_0^tb^2(X_s)e^{-2\int_s^ta(X_r)\,dr}\,ds$$ 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 in Markovian environments. The time asymptotic behaviors are naturally expressed in terms of solutions to the well-studied fixed-point equation in law $$X\overset{d}{=}AX+B$$ with $$X\perp \!\!\!\!\perp (A,B)$$.

### MSC:

 60J60 Diffusion processes 60J27 Continuous-time Markov processes on discrete state spaces

MCQueue
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### References:

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