Infinite horizon optimization for finite state Markov chain

We consider the infinite horizon optimal control of a finite state Markov chain from the point of view of overtaking optimality and the long-run average cost. The stochastic model is cast into a deterministic framework by considering the distribution of the original state as a new state. We characterize and prove existence of stationary strategies which have minimal cost growth rate in the class of all nonanticipative strategies. Restricting attention only to stationary strategies we show that for every given initial state there exists an overtaking optimal strategy. Finally, under more restrictive conditions, we establish the existence of a stationary overtaking optimal strategy for all the initial conditions.