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 [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].

Keywords:

• Optimal stopping
• Prophet inequality
• Random horizon
• AMS 2000 Subject Classifications: 60G40
• 62L15

Acknowledgment

The author thanks two anonymous referees for several helpful suggestions.

Notes

Email: [email protected]

Additional information

Notes on contributors

Pieter C. Allaart *

Email: [email protected]