Moderate deviation principles for stochastic differential equations in fast-varying Markovian environment

In this paper, we proved moderate deviation principles for a fully coupled two-time-scale stochastic systems, where the slow process is given by stochastic differential equations with small noise, while the fast process is a rapidly changing purely jump process on finite state space. The system is fully coupled in that the drift and diffusion coefficients of the slow process, as well as the jump distribution of the fast process, depend on states of both processes. Moreover, the diffusion component in the slow process can be degenerate, and the drift is allowed to have polynomial growth. Our approach is based on the combination of the weak convergence method from [A. Budhiraja, P. Dupuis, and A. Ganguly, Electron. J. Probab. 23 (2018), pp. 1-33; Ann. Probab. 44 (2016), pp. 1723-1775] with Poisson equation for the fast-varying purely jump process.