On Elfving problem

Abstract

In this paper we analyze the optimal stopping time problem known as Elfving problem. The problem can be formulated as follows: a decision maker observes offers which appear at jump times of a Poisson process. The decision concerning acceptance or rejection of a presented offer is made at the moment of its appearance. Once rejected, offer cannot be considered again. A reward for the decision maker is equal to the discounting value of the selected offer. The aim of the decision maker is to maximize his expected reward. In this paper we present applications of the described model in economy and sociology. We also present the optimal strategy and a differential equation which allows us to compute the optimal mean reward. Moreover, we analyze the influence of model’s parameters on the value of optimal mean reward and we compute the mean time of waiting for the optimal stopping time.