Singularly Perturbed Switching Diffusions: Rapid Switchings and Fast Diffusions

Motivated by many problems in optimization and control, this paper is concerned with singularly perturbed systems involving both diffusions and pure jump processes. Two models are treated. In the first model, the jump process changes very rapidly by comparison with the diffusion processes. In the second model, the diffusions change rapidly in comparison with the jump process. Asymptotic expansions are developed for the transition density vectors via a constructive method; justification of the asymptotic expansions and analysis of the remainders are provided.