On the Optimality of Structured Policies in Countable Stage Decision Processes, III: Regular Terminal Value Functions

The study of structured countable stage decision processes is continued. Rather than requiring the terminal value functions to be what are called regular-structured [6], we only require them to be regular. Since the infinite horizon values are defined as limits of the finite horizon values as the horizon length goes to infinity, the terminal value functions play an important role. In applications, regular value functions often satisfy certain natural bounds, whereas the structured ones often have additional properties. Thus our conditions broaden the scope of well defined decision processes. Furthermore, we show that the optimal value functions are the unique regular value functions to satisfy the optimality equations. Thus, the computational method of seeking a solution of the optimality equations is strengthened. We also present conditions which guarantee that there exists an optimal strategy which has additional structure not easily established using ordinary means. We illustrate our results with an application to an optimal dividend and liquidation policy model.