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Calculating optimal policies is known to be computationally difficult for Markov decision processes with Borel state and action spaces and for partially observed Markov decision processes even with finite state and action spaces. This paper studies finite-state approximations of discrete time Markov decision processes with Borel state and action spaces, for both discounted and average costs criteria. The stationary policies thus obtained are shown to approximate the optimal stationary policy with arbitrary precision under mild technical conditions. For compact-state MDPs, we obtain explicit rates of convergence bounds quantifying how the approximation improves as the size of the approximating finite state space increases. Using information theoretic arguments, the order optimality of the obtained rates of convergence is established for a large class of problems. We also show that, as a pre-processing setup, action space can taken to be finite with sufficiently large number points for the finite-state approximation problem; thereby, well known algorithms, such as value or policy iteration, Q-learning, etc., can be used to calculate near optimal policies.

On the Asymptotic Optimality of Finite Approximations to Markov Decision Processes with Borel Spaces