This work develops asymptotic expansions of systems of partial differential equations associated with multi-scale switching diffusions. The switching process is modeled by using an inhomogeneous continuous- time Marko...This work develops asymptotic expansions of systems of partial differential equations associated with multi-scale switching diffusions. The switching process is modeled by using an inhomogeneous continuous- time Markov chain. In the model, there are two small parameters ε and δ. The first one highlights the fast switching, whereas the other delineates the slow diffusion. Assuming that ε and δ are related in that ε = δγ, our results reveal that different values of γ lead to different behaviors of the underlying systems, resulting in different asymptotic expansions. Although our motivation comes from stochastic problems, the approach is mainly analytic and is constructive. The asymptotic series are rigorously justified with error bounds provided. An example is provided to demonstrate the results.展开更多
基金supported in part by the Air Force Office of Scientific Research under FA9550-15-1-0131
文摘This work develops asymptotic expansions of systems of partial differential equations associated with multi-scale switching diffusions. The switching process is modeled by using an inhomogeneous continuous- time Markov chain. In the model, there are two small parameters ε and δ. The first one highlights the fast switching, whereas the other delineates the slow diffusion. Assuming that ε and δ are related in that ε = δγ, our results reveal that different values of γ lead to different behaviors of the underlying systems, resulting in different asymptotic expansions. Although our motivation comes from stochastic problems, the approach is mainly analytic and is constructive. The asymptotic series are rigorously justified with error bounds provided. An example is provided to demonstrate the results.
