An Optimal Stopping Rule for Approaching a Border that Should not be Crossed

摘要

We consider an optimal stopping problem with some of the features of the blackjack type games. Let X_1, X_2,…,X_N be a finite sequence of nonnegative random variables. A decision maker observes sequentially the values and decides whether to stop or to continue. If he decides to stop at the moment k he obtains a payoff dependent on the sum X_1+…+X_k. The greater the sum, the more the decision maker gains, unless the sum exceeds a positive number T - a limit given in the problem. If so, the decision maker loses all or part of his payoff. A special case of such a problem is considered in details. In this case a simple optimal stopping rule is found. Some examples and practical questions are discussed as well.