Markov decision processes approximation with coupled dynamics via Markov deterministic control systems

This article presents an approximation of discrete Markov decision processes with small noise on Borel spaces with an infinite horizon and an expected total discounted cost by the corresponding deterministic Markov process. In both cases, the dynamics evolve through a system consisting of two coupled difference equations. It is assumed that the difference equations of the system are perturbed by a small noise. Under our assumptions, a bound for the stability index is given, and the optimal cost convergence rate is estimated using a small perturbation parameter. Moreover, the convergence of the optimal policy on compact subsets is verified. Finally, two examples are presented to illustrate the developed theory.