An application of prophet regions to optimal stopping with a random number of observations

Abstract

Let X 1 , X 2 , . . . be any sequence of nonnegative integrable random variables, and let N ∈ {1, 2, . . . } be a random variable with known distribution, independent of X 1 , X 2 , . . . . The optimal stopping value sup t E(X t I(N ≥ t)) is considered for two players: one who has advance knowledge of the value of N , and another who does not. Sharp ratio and difference inequalities relating the two players' optimal values are given in a number of settings. The key to the proofs is an application of a prophet region for arbitrarily dependent random variables by Hill and Kertz (Trans. Amer. Math. Soc. 278, 197-207 (1983)).AMS 2000 subject classification: 60G40, 62L15.