On Finite Stage Decision Processes

We develop finite stage decision processes in the notation of Denardo [3] and provide conditions sufficient to prove that (i) n-stage returns exist (are unique) and are “subfinite”, (ii) optimal returns exist and are finite, (iii) Bellman’s [I] finite stage “optimality equations” and “principle of optimality” hold, (iv) there exist e-optimal and/or optimal strategies, (v) and there exist e-optimal and/or optimal “structured” strategies. We show how known results regarding (i) the optimality of Bore1 measurable policies in a certain Markov decision process, and (ii) the optimality of generalized (s,S) policies in a certain inventory process, fall into our framework.