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Predicting the Supremum: Optimality of ‘Stop at Once or Not at All’

Part of: Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Pieter C. Allaart
Pieter C. 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: allaart@unt.edu
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Abstract

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Let (Xt)0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (Xt) ‘as close as possible’ to its eventual supremum MT := sup0 ≤ t ≤ TXt, when the reward for stopping at time τ ≤ T is a nonincreasing convex function of MT - Xτ. Under fairly general conditions on the process (Xt), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.

Keywords

Random walkLévy processoptimal predictionultimate supremumstopping timeskew symmetryconvex function

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G50: Sums of independent random variables; random walks
Secondary: 60G25: Prediction theory
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 3 , September 2012 , pp. 806 - 820
Copyright
© Applied Probability Trust 

Footnotes

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

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