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Optimal Stopping with Rank-Dependent Loss

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Alexander V. Gnedin
Alexander V. Gnedin*
Affiliation:
Utrecht University
*
∗Postal address: Department of Mathematics, Utrecht University, Postbus 80010, 3508 TA Utrecht, The Netherlands. Email address: [email protected]
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Abstract

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For τ, a stopping rule adapted to a sequence of n independent and identically distributed observations, we define the loss to be E[q(Rτ)], where Rj is the rank of the jth observation and q is a nondecreasing function of the rank. This setting covers both the best-choice problem, with q(r) = 1(r > 1), and Robbins' problem, with q(r) = r. As n tends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.

Keywords

Optimal stoppingRobbins' problembest-choice problemplanar Poisson process

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 996 - 1011
Copyright
Copyright © Applied Probability Trust 2007 

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