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Stopping the maximum of a correlated random walk, with cost for observation

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Pieter Allaart
Pieter Allaart*
Affiliation:
University of North Texas
*
∗ Postal address: Mathematics Department, University of North Texas, PO Box 311430, Denton, TX 76203-1430, USA. Email address: [email protected]
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Abstract

Let (S n ) n≥0 be a correlated random walk on the integers, let M 0 ≥ S 0 be an arbitrary integer, and let M n = max{M 0, S 1,…, S n }. An optimal stopping rule is derived for the sequence M n - nc, where c > 0 is a fixed cost. The optimal rule is shown to be of threshold type: stop at the first time that M n - S n ≥ Δ, where Δ is a certain nonnegative integer. An explicit expression for this optimal threshold is given.

Keywords

Correlated random walkmomentumstopping ruleoptimality principlelinear difference equation

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 998 - 1007
Copyright
Copyright © Applied Probability Trust 2004 

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