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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: mueller@wior.uni-karlsruhe.de
∗∗ 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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