| Motivated by many applications arising from control and optimization problems, this dissertation is concerned with singularly perturbed switching diffusions with fast and slow components. The diffusions involved are multidimensional, which evolve in an order of magnitude slower than that of the jumps or the switchings. The jump component may be decomposed into several weakly irreducible classes or groups of recurrent states. There are rapid switchings in each of such groups, whereas the jumps among different groups take place much less frequently. Our study focuses on solutions of the associated Cauchy problem for the forward equation or Kolmogorov-Fokker-Planck equation via singular perturbation techniques. Recurrent states and inclusion of absorbing states are the two considered cases. Under simple conditions, we construct the outer-inner expansions and justify that the expansions are valid with desired errors under the uniform topology. Such results are very useful for obtaining asymptotic properties of an aggregated process (aggregation with respect to each recurrent group) and for getting asymptotic optimal controls of switching diffusion systems involving two-time scale. |
