Probability Maximizing Approach to Optimal Stopping of Random Fields
The paper deals with a problem of optimal stopping for non-negative random fields. As in Mandelbaum and Vanderbei (1981) we restrict ourselves to the class of predictable stopping points. A necessary and sufficient condition for existence an optimal strategy is given. The result is used to solve the probability maximizing version of the problem of optimal stopping for random fields indexed by a countable partially ordered set. It generalizes the probability maximizing approach to the problem of optimal stopping for stochastic processes formulated by Bojdecki (1978). We specialize our results to the problem of optimal stopping for several Markov chains. Examples concerning the problem of optimal allocation of different treatments and some problems of optimal selection are given.
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