Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-27T22:08:31.775Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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’.

References

Allaart, P. C. (2010). A general ‘bang–bang’ principle for predicting the maximum of a random walk. J. Appl. Prob. 47, 1072–1083.CrossRefGoogle Scholar
Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press.Google Scholar
Bernyk, V., Dalang, R. C. and Peskir, G. (2008). The law of the supremum of a stable Lévy process with no negative Jumps. Ann. Prob. 36, 1777–1789.Google Scholar
Bernyk, V., Dalang, R. C. and Peskir, G. (2011). Predicting the ultimate supremum of a stable Lévy process with no negative Jumps. Ann. Prob. 39, 2385–2423.Google Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–332.Google Scholar
Du Toit, J. and Peskir, G. (2007). The trap of complacency in predicting the maximum. Ann. Prob. 35, 340–365.CrossRefGoogle Scholar
Du Toit, J. and Peskir, G. (2009). Selling a stock at the ultimate maximum. Ann. Appl. Prob. 19, 983–1014.Google Scholar
Graversen, S. E., Peskir, G. and Shiryaev, A. N. (2000). Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Theory Prob. Appl. 45, 41–50.Google Scholar
Hlynka, M. and Sheahan, J. N. (1988). The secretary problem for a random walk. Stoch. Process. Appl. 28, 317–325.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Pedersen, J. L. (2003). Optimal prediction of the ultimate maximum of Brownian motion. Stoch. Stoch. Reports 75, 205–219.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Shiryaev, A., Xu, Z. and Zhou, X. Y. (2008). Thou shalt buy and hold. Quant. Finance 8, 765–776.Google Scholar
Yam, S. C. P., Yung, S. P. and Zhou, W. (2009). Two rationales behind ‘buy-and-hold or sell-at-once’ strategy. J. Appl. Prob. 46, 651–668.Google Scholar