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On the optimal stopping values induced by general dependence structures

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  14 July 2016

Alfred Müller  and
Ludger Rüschendorf
Alfred Müller*
Affiliation:
Universität Karlsruhe
Ludger Rüschendorf*
Affiliation:
Universität Freiburg
*
∗ Postal address: Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Geb. 20.21, D-76128 Karlsruhe, Germany. Email address: [email protected]
∗∗ Postal address: Institut für Mathematische Stochastik, Universität Freiburg, Eckerstrasse 1, D-79104 Freiburg, Germany.
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Abstract

The optimal stopping value of random variables X1,…,Xn depends on the joint distribution function of the random variables and hence on their marginals as well as on their dependence structure. The maximal and minimal values of the optimal stopping problem is determined within the class of all joint distributions with fixed marginals F1,…,Fn. They correspond to some sort of strong negative or positive dependence of the random variables. Any value inbetween these two extremes is attained for some dependence structures. The determination of the minimal value is based on some new ordering results for probability measures, in particular on lattice properties of stochastic orderings. We also identify properties of dependence structures leading to the minimal optimal stopping value. In the proofs we need an extension of Strassen's theorem on representation of the convex order which reveals that convex ordered distributions can be coupled by a two-step martingale (X,Y) with the additional property that Y is stochastically increasing in X.

Keywords

Optimal stoppingdistributions with given marginalsmaximal dependencestochastic orderingextension of Strassen's theorem

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 62L15: Optimal stopping
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 672 - 684
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Barlow, R. E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Blackwell, D. (1953). Equivalent comparisons of experiments. Ann. Math. Statist./ 24, 265–272.Google Scholar
Chow, Y. S., Robbins, H., and Siegmund, D. (1971). Great Expectations: Theory of Optimal Stopping. Houghton Mifflin, Boston.Google Scholar
Elton, J., and Hill, T. P. (1992). Fusions of a probability distribution. Ann. Prob./ 20, 421–454.Google Scholar
Elton, J., and Hill, T. P. (1998). On the basic representation theorem for convex domination of measures. J. Math. Anal. Appl./ 228, 449–466.CrossRefGoogle Scholar
Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality. Springer, New York.Google Scholar
Kellerer, H. G. (1972). Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. 198, 99–122.Google Scholar
Kertz, R. P. and Rösler, U. (1992). Martingales with given maxima and terminal distributions. Israel J. Math. 69, 173–192.Google Scholar
Lai, T. L., and Robbins, H. (1976). Maximally dependent random variables. Proc. Nat. Acad. Sci. USA 73, 286–288.Google Scholar
Lai, T. L., and Robbins, H. (1978). A class of dependent random variables and their maxima. Z. Wahrscheinlichkeitsth. 42, 89–112.Google Scholar
Machina, M., and Pratt, J. (1997). Increasing risk: some direct constructions. J. Risk Uncertainty 14, 103–127.Google Scholar
Makowski, A. M. (1994). On an elementary characterization of the increasing convex ordering, with an application. J. Appl. Prob. 31, 834–840.Google Scholar
Müller, A. (1996). Orderings of risks: a comparative study via stop-loss transforms. Insurance Math. Econom. 17, 215–222.Google Scholar
Müller, A. (1998). Comparing risks with unbounded distributions. J. Math. Econom. 30, 229–239.Google Scholar
Müller, A. (2001). Bounds for optimal stopping values of dependent random variables with given marginals. Statist. Prob. Lett. 52, 73–78.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems, Vol. 1, Theory. Springer, New York.Google Scholar
Rinott, Y., and Samuel-Cahn, E. (1987). Comparisons of optimal stopping values and prophet inequalities for negatively dependent random variables. Ann. Statist. 15, 1482–1490.Google Scholar
Rinott, Y., and Samuel-Cahn, E. (1991). Orderings of optimal stopping values and prophet inequalities for certain multivariate distributions. J. Multivariate Anal. 37, 104–114.Google Scholar
Rothschild, M., and Stiglitz, J. E. (1970). Increasing risk. I. A definition. J. Econom. Theory 2, 225–243.Google Scholar
Rüschendorf, L. (1980). Inequalities for the expectation of Δ-monotone functions. Z. Wahrscheinlichkeitsth. 54, 341–349.CrossRefGoogle Scholar
Rüschendorf, L. (1981). Stochastically ordered distributions and monotonicity of the OC-function of sequential probability ratio test. Math. Operationsforsch. Statist. Ser. Optimization 12, 327–338.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and their Applications. Academic Press, London.Google Scholar
Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, Chichester.Google Scholar
Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.Google Scholar