2019/04/02 by Lindskog, Filip, Majumder, Abhishek Pal
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1904.01474
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 a countable 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 ∫0tb2(Xs)e^-2∫sta(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 in Markovian environments. Exact long time behaviors are naturally expressed in terms of solutions to the well-studied fixed-point equation in law X\stackreld=AX+B with X \indep (A,B).