Multiple-stopping problems with random horizon

Abstract

An optimal multiple-stopping problem with a random horizon and a general structure of rewards is considered. The problem can be formulated as follows. A decision maker has commodities of the same type for sale. Therefore, he can accept at most offers, which he observes sequentially. The decision on acceptance or rejection of the offer must be made on the basis of the past and current observations. Each accepted offer brings in a profit of if the offer is accepted before a random horizon and 0 otherwise. That means that after time the commodities cannot be sold. The aim of the decision-maker is to maximize the expected total profit. An optimal selection strategy is obtained by applying a general theorem which shows that a model with random horizon can be reduced to a model with a discount function. A case with random horizon independent of time and magnitude of offers is analysed in detail. As applications, two problems are solved: a multiple-stopping problem with a random horizon, a discount function and a Poisson stream of offers and a ‘timeless’ multiple-stopping problem with a random number of offers and a discount factor.