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A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Pieter Allaart
Pieter Allaart*
Affiliation:
University of North Texas
*
∗Postal address: Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203-5017, USA. Email address: [email protected]
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Abstract

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Let (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (Bt) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

Keywords

Bernoulli random walkBrownian motionoptimal predictionultimate maximumstopping timeconvex function

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G50: Sums of independent random variables; random walks 60J65: Brownian motion 60G25: Prediction theory
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1072 - 1083
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Supported in part by the Japanese GCOE Program G08: ‘Fostering Top Leaders in Mathematics-Broadening the Core and Exploring New Ground’.

References

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