Z. Wahrscheinlichkeitstheorie verw. Gebiete 57, 253-264 (1981) Zeitschrift flir Wahrscheinlichkeitstheorie und verwandte Gebiete © Springer-Verlag 1981 Optimal Stopping and Supermartingales over Partially Ordered Sets A. Mandelbaum 1 and R.J. Vanderbei 2 1 Dept. of Operations Research, Cornell University, Ithaca, N.Y. 14853, USA 2 Dept. of Mathematics, Cornell University, Ithaca, N,y' 14853, USA 1. Introduction 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 pro- cesses. In 1953, Snell [12J discovered the relation between optimal stopping of a random sequence and supermartingales. In 1963, Dynkin [3] described the op- timal stopping rule for a Markov process in terms of excessive functions. In 1966, Haggstrom [7J, motivated by problems of sequential experimental de- sign, 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 indepen- dent 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 cer- tain subclass which we call predictable. An important result is that the super- martingale 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 con- straints and Walsh's theory of multiharmonic functions (Sect. 4). 1.2. Let Zt be a sequence of random variables adapted to a filtration $',. The classical optimal stopping problem is to find a stopping time T* which is op- Walsh [15] has extended this to the continuous two parameter case 0044-3719/81/005 7/025 3/$02.40