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 \(X\perp \!\!\!\!\perp (A,B)\).


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


Full Text: DOI arXiv Euclid


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