Markov control processes with pathwise constraints

This paper deals with discrete-time Markov control processes in Borel spaces, with unbounded rewards. The criterion to be optimized is a long-run sample-path (or pathwise) average reward subject to constraints on a long-run pathwise average cost. To study this pathwise problem, we give conditions for the existence of optimal policies for the problem with ‘expected’ constraints. Moreover, we show that the expected case can be solved by means of a parametric family of optimality equations. These results are then extended to the problem with pathwise constraints.