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November 2020 Exact long time behavior of some regime switching stochastic processes
Filip Lindskog, Abhishek Pal Majumder
Bernoulli 26(4): 2572-2604 (November 2020). DOI: 10.3150/20-BEJ1196

Abstract

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 $\int _{0}^{t}b^{2}(X_{s})e^{-2\int _{s}^{t}a(X_{r})\,dr}\,ds$ 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 $X\stackrel{d}{=}AX+B$ with $X\perp \!\!\!\!\perp (A,B)$.

Citation

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Filip Lindskog. Abhishek Pal Majumder. "Exact long time behavior of some regime switching stochastic processes." Bernoulli 26 (4) 2572 - 2604, November 2020. https://doi.org/10.3150/20-BEJ1196

Information

Received: 1 April 2019; Revised: 1 October 2019; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256153
MathSciNet: MR4140522
Digital Object Identifier: 10.3150/20-BEJ1196

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 4 • November 2020
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