This paper studies the average cost for controlled systems given as solutions of the martingale problem for their generator. The control problem is reformulated as a linear programming problem and conditions are given for the existence of an optimal solution. It is further shown that the optimal control can be taken to depend only on the history of the system and that this cost remains optimal for systems with different initial distributions.
Richard H. Stockbridge. "Time-Average Control of Martingale Problems: A Linear Programming Formulation." Ann. Probab. 18 (1) 206 - 217, January, 1990. https://doi.org/10.1214/aop/1176990945