  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).

Probability