Large deviations for Markov-modulated diffusion processes with rapid switching

TL;DR

This paper establishes a large deviations principle for Markov-modulated diffusion processes with rapid switching, providing a rigorous mathematical framework for understanding their small noise asymptotics in such regimes.

Contribution

It proves the joint large deviations principle for the process and the occupation measure of the Markov chain under rapid switching conditions.

Findings

Established the joint large deviations principle for the process and occupation measure.

Proved exponential tightness and local large deviations bounds.

Applied contraction principle to derive separate large deviations results.

Abstract

In this paper, we study small noise asymptotics of Markov-modulated diffusion processes in the regime that the modulating Markov chain is rapidly switching. We prove the joint sample-path large deviations principle for the Markov-modulated diffusion process and the occupation measure of the Markov chain (which evidently also yields the large deviations principle for each of them separately by applying the contraction principle). The structure of the proof is such that we first prove exponential tightness, and then establish a local large deviations principle (where the latter part is split into proving the corresponding upper bound and lower bound).