An application of prophet regions to optimal stopping with a random number of observations - Allaart, Pieter C. (Optimization, 2004)
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 [T.P. Hill and R.P. Kertz (1983). Stop rule inequalities for uniformly bounded sequences of random variables. Trans. Amer. Math. Soc., 278, 197-207].