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

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 o...

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Bibliographic Details
Published inOptimization Vol. 53; no. 4; pp. 331 - 338
Main Author Allaart, Pieter C.
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis Group 01.08.2004
Taylor & Francis LLC
Subjects
AMS 2000 Subject Classifications: 60G40
Game theory
Mathematical models
Optimal stopping
Optimization
Prophet inequality
Random horizon
Random variables
Studies
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ISSN0233-1934
1029-4945
DOI10.1080/02331930410001716829

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Summary: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].
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ISSN:0233-1934
1029-4945
DOI:10.1080/02331930410001716829