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Optimal stopping with discount and observation costs

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  14 July 2016

Robert Kühne  and
Ludger Rüschendorf
Robert Kühne*
Affiliation:
University of Freiburg
Ludger Rüschendorf*
Affiliation:
University of Freiburg
*
∗Postal address: Institut für Mathematische Stochastik, Universität Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany
∗Postal address: Institut für Mathematische Stochastik, Universität Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany
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Abstract

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

Keywords

Optimal stoppingmax-stable distributionsPoisson processes

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 64 - 72
Copyright
Copyright © 2000 by The Applied Probability Trust 

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References

Bruss, F. T., and Rogers, L. C. G. (1991). Embedding optimal selection problems in a Poisson process. Stoch. Proc. Appl. 38, 267–278.CrossRefGoogle Scholar
Chow, Y. S., Robbins, R., and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, New York.Google Scholar
Elfving, G. (1967). A persistence problem connected with a point process. J. Appl. Prob. 4, 77–89.CrossRefGoogle Scholar
Flatau, J., and Irle, A. (1984). Optimal stopping for extremal processes. Stoch. Proc. Appl. 16, 99–111.Google Scholar
Gnedin, A. (1996). On the full information best-choice problem. J. Appl. Prob. 33, 678–687.Google Scholar
de Haan, I., and Verkade, E. (1987). On extreme value theory in the presence of a trend. J. Appl. Prob. 24, 62–76.Google Scholar
Kennedy, D. P., and Kertz, R. P. (1990). Limit theorems for threshold stopped random variables. Adv. Appl. Prob. 22, 396–411.Google Scholar
Kennedy, D. P., and Kertz, R. P. (1991). The asymptotic behaviour of the reward sequence in the optimal stopping of iid random variables. Ann. Prob. 19, 329–341.Google Scholar
Kennedy, D. P., and Kertz, R. P. (1992). Limit theorems for suprema. Threshold stopped r.v.s with costs and discounting, with applications to optimal stopping. Adv. Appl. Prob. 22, 241–266.Google Scholar
Kühne, R. (1997). Probleme des asymptotisch optimalen Stoppens. Dissertation, Universität Freiburg.Google Scholar
Kühne, R. and Rüschendorf, L. (1998). Approximation of optimal stopping problems. Preprint 32, Mathematische Fakultät, Universität Freiburg.Google Scholar
Resnick, S. I. (1987). Extreme Values. Regular Variation and Point Processes. Springer, New York.Google Scholar