Diffusion approximation and stability of stochastic differential equations with singular perturbation

This paper investigates diffusion approximation and stability of non-autonomous singularly perturbed stochastic differential equations with locally Lipschitz continuous coefficients. By using the first-order perturbation test function method and formulation of the martingale problem, the averaging principle is established and the averaging system is obtained. Under appropriate conditions, if the averaging system is exponentially stable, this paper shows that the original slow component is also uniformly asymptotically stable. Since the averaging system is often simpler than the original system, this stability result is interesting. Finally, several examples illustrate our results.