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Prophet region for independent random variables with a discount factor

By Frans Boshuizen

Abstract

AbstractLet X1, …, Xn be a sequence of independent [0, 1]-valued random variables and let 0 < β ≤ 1. A complete comparison is made between the optimal stopping value V(X1, βX2, …, βn−1Xn) = sup{Eβτ−1Xτ: τ is a stop rule for X1, …, Xn} and E(max1≤i≤nβi−1Xi). In fact it is shown that the set {(x,y): x = V(X1, βX2, …, βn−1Xn), y = E(max1≤i≤nβi−1Xi), some sequence of independent [0, 1]-valued random variables X1, …, Xn} is precisely the set {(x, y): x ≤ y ≤ Ψβ(x), 0 ≤ x ≤ 1}, where Ψβ(x) = 2x − x2β if x ∈ [0, 1 − √1−β] and Ψβ(x) = 1 − 2((√1−β − (1 − β))β)(1 − x) if x ∈ [1 − √1−β, 1]

Publisher: Published by Elsevier Inc.
Year: 1991
DOI identifier: 10.1016/0047-259X(91)90112-F
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