Optimal Stopping and Supermartingales over Partially Ordered Sets

1.1. The subject of this paper is the problem of optimal stopping for discrete multiparameter stochastic processes; in particular, for a family of Markov processes. In 1953, Snell [12J discovered the relation between optimal stopping of a random sequence and supermartingales. In 1963, Dynkin [3] described the optimal stopping rule for a Markov process in terms of excessive functions. In 1966, Haggstrom [7J, motivated by problems of sequential experimental design, extended Snell's results to processes indexed by a tree. All these results are particular cases of the general theory of optimal stopping for a family of random variables indexed by a partially ordered set. This general setting was considered by Krengel and Sucheston [8] who proved a number of general theorems and applied them to the case of functionals of a family of independent identically distributed random variables. We start by discussing, in the spirit of Snell's theory, the general optimal stopping problem over a partially ordered set. The proofs of the main theorems are similar to Haggstrom's proofs but for completeness we outline them briefly in Sect. 6. Our emphasis is on the nature of stopping points taking values in partially ordered sets. Not all stopping points are appropriate but only a certain subclass which we call predictable. An important result is that the supermartingale sampling theorem holds for predictable stopping points. 1 The general theory is applied to a family of Markov chains and we get results analogous to Dynkin's results. An example for two independent random walks is considered in Sect. 3. Finally, we discuss the relation between optimal stopping with time constraints and Walsh's theory of multiharmonic functions (Sect. 4).