Abhishek Pal Majumder: Long time asymptotics of some regime switching processes

Abstract: Regime switching stochastic processes are indispensable in modelling several contexts when underlying model parameters are observed to fluctuate over time in a Markovian manner. In diffusion context we study one such example: Ornstein-Uhlenbeck process when the drift and diffusion coefficients switch through Markovian regimes through functional formsrespectively denoted by \(a(\cdot),b(\cdot)\). The switching dynamics is denoted by process \(X_{\cdot}\). Exact long time asymptotics are established in different cases (positive recurrent, null recurrent and transient) that are determined by signs of the expected drift under stationarity of the underlying regime process \(X_{\cdot}\). Previously only different types of tail behaviours of stationary distribution were investigated under stable regime \(E_{\pi}a(\cdot) >0\) while here we get an explicit​ characterization in distribution. Additionally time limit results for a stochastic integral of specific form \(\mathcal{I}_{t}:=\int_{0}^{t}b^{2}\big(X_{s}\big)e^{-2\int_{s}^{t}a(X_{s_{1}})ds_{1}}ds\) is established in three different cases which is crucial in deriving precise long time descriptions of the deterministic SIS epidemic model under Markovian switching environments.​