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

Let X1,X2,... be any sequence of nonnegative integrable random vari-ables, and let N ∈ {1, 2,...} be a random variable with known distribution, independent of X1,X2,.... The optimal stopping value suptE(XtI(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 rando