Averaging and large deviation principles for two time-scale regime-switching processes

Asymptotic behavior of a fully coupled two time-scale system is investigated, whose slow process is a diffusion process and fast process is a purely jumping process on a discrete state space. As the fast process is allowed to be on an infinitely countable state space, in order to establish the averaging principle, we impose separately the strongly ergodic condition and a weaker one on the Markov process associated with the fast component with each fixed slow component. We showed that under strongly ergodic condition, the limit system, being characterized by an ordinary differential equation, admits a unique solution, and the slow process converges in the L1-norm to the limit system. However, under weaker ergodic condition, the limit system admits a solution, but not necessarily unique, and the slow process converges weakly to a solution of the limit system. Besides, we propose a coupling method for jumping processes to decouple the interaction between the slow component and fast component based on Skorokhod's representation theorem for jumping processes. Furthermore, the large deviation principle is established for the two time-scale system using the nonlinear semigroup approach under two different slow-fast time-scale ratios. Different types of drift conditions are proposed when the state space of the fast jumping process is infinitely countable.