 
Summary: An application of prophet regions to optimal
stopping with a random number of observations
Pieter C. Allaart
University of North Texas
April 6, 2004
Abstract
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 supt E(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 random
variables by Hill and Kertz (Trans. Amer. Math. Soc. 278, 197207 (1983)).
AMS 2000 subject classification: 60G40, 62L15.
Key words: Optimal stopping, prophet inequality, random horizon
Author's address: Mathematics Department, University of North Texas, P.O. Box 311430,
Denton, TX 762031430; email: allaart@unt.edu
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