Asymptotic Optimality and Rates of Convergence of Quantized Stationary Policies in Stochastic Control

We consider the discrete approximation of stationary policies for a discrete-time Markov decision process with Polish state and action spaces under total, discounted, and average cost criteria. Deterministic stationary quantizer policies are introduced and shown to be able to approximate optimal deterministic stationary policies with arbitrary precision under mild technical conditions, thus demonstrating that one can search for ε-optimal policies within the class of quantized control policies. We also derive explicit bounds on the approximation error in terms of the quantization rate.