SINGULARLY PERTURBED MULTI-SCALE SWITCHING DIFFUSIONS

This work is concerned with singularly perturbed multi-scale switching diffusions. The switching process is a two-time-scale Markov chain with slow and fast components subject to weak and strong interactions. In the model, there are two small parameters ε and δ. The first one highlights the fast changing part of the switching process, and the other delineates the slow diffusion. We treat the case that ε and δ are related in that ε = δ γ . Under certain conditions, asymptotic expansions of the probability densities for the underlying processes are developed. The approach is constructive and the asymptotic series are rigorously justified with error bounds.