Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T00:23:11.037Z Has data issue: false hasContentIssue false
  • English
  • Français

On approximative solutions of optimal stopping problems

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  01 July 2016

Andreas Faller  and
Ludger Rüschendorf
Andreas Faller*
Affiliation:
University of Freiburg
Ludger Rüschendorf*
Affiliation:
University of Freiburg
*
∗ Andreas Faller died unexpectedly on 30 June 2011.
∗∗ Postal address: Mathematical Stochastics, University of Freiburg, Eckerstr. 1, 79104 Freiburg, Germany. Email address: ruschen@stochastik.uni-freiburg.de
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.

In this paper we establish an extension of the method of approximating optimal discrete-time stopping problems by related limiting stopping problems for Poisson-type processes. This extension allows us to apply this method to a larger class of examples, such as those arising, for example, from point process convergence results in extreme value theory. Furthermore, we develop new classes of solutions of the differential equations which characterize optimal threshold functions. As a particular application, we give a fairly complete discussion of the approximative optimal stopping behavior of independent and identically distributed sequences with discount and observation costs.

Keywords

Optimal stoppingmax-stable distributionPoisson process

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 1086 - 1108
Copyright
Copyright © Applied Probability Trust 2011 

References

Bruss, F. T. and Rogers, L. C. G. (1991). Embedding optimal selection problems in a Poisson process. Stoch. Process. Appl. 38, 267–278.CrossRefGoogle Scholar
Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston, MA.Google Scholar
De Haan, L. and Verkaade, E. (1987). On extreme-value theory in the presence of a trend. J. Appl. Prob. 24, 62–76.CrossRefGoogle Scholar
Faller, A. (2009). Approximative Lösungen von Mehrfachstoppproblemen. , University of Freiburg.Google Scholar
Faller, A. and Rüschendorf, L. (2011). On approximative solutions of multistopping problems. Ann. Appl. Prob. 21, 1965–1965.CrossRefGoogle Scholar
Gnedin, A. V. and Sakaguchi, M. (1992). On a best choice problem related to the Poisson process. In Strategies for Sequential Search and Selection in Real Time (Amherst, MA, 1990; Contemp. Math. 125), American Mathematical Society, Providence, RI, pp. 59–64.CrossRefGoogle Scholar
Kennedy, D. P. and Kertz, R. P. (1990). Limit theorems for threshold-stopped random variables with applications to optimal stopping. Adv. Appl. Prob. 22, 396–411.CrossRefGoogle Scholar
Kennedy, D. P. and Kertz, R. P. (1991). The asymptotic behavior of the reward sequence in the optimal stopping of i.i.d. random variables. Ann. Prob. 19, 329–341.CrossRefGoogle Scholar
Kühne, R. and Rüschendorf, L. (2000a). Approximation of optimal stopping problems. Stoch. Process. Appl. 90, 301–325.CrossRefGoogle Scholar
Kühne, R. and Rüschendorf, L. (2000b). Optimal stopping with discount and observation costs. J. Appl. Prob. 37, 64–72.CrossRefGoogle Scholar
Kühne, R. and Rüschendorf, L. (2003). Optimal stopping and cluster point processes. Statist. Decisions 21, 261–282.CrossRefGoogle Scholar
Kühne, R. and Rüschendorf, L. (2004). Approximate optimal stopping of dependent sequences. Theory Prob. Appl. 48, 465–480.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar