Singularly Perturbed Markov Chains with Two Small Parameters: A Matched Asymptotic Expansion

This work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed Markov chains with two small parameters. It is motivated by the model of a cost-minimizing firm involving production planning and capacity expansion and a two-level hierarchical decomposition. Our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation. Different from previous work on singularly perturbed Markov chains, the inner expansion terms are constructed by solving certain partial differential equations. The methods of undetermined coefficients are used. The error bound is obtained.