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SPEPS Preprint

Author(s): Svetlozar T. Rachev, University of California and University of Karlsruhe
Werner Römisch, Humboldt-University Berlin
Title: Quantitative stability in stochastic programming – The method of probability metrics
Date of Acceptance: 20.12.2000
Submission Date: 04.12.2000
Series Title: Stochastic Programming E-Print Series
(SPEPS)
Editors: Julie L. Higle; Werner Römisch; Surrajeet Sen
Keywords (eng): quantitative stability, Stochastic programming, probability metrics, Fortet-Mourier metrics, empirical approximations, two-stage models, chance constrained models, stable portfolio models
Appeared in: Mathematics of operations research 4 (Vol. 27, 2002)
Institute for Operations Research and the Management Sciences (Linthicum, Md.)
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Abstract (eng):
Quantitative stability of optimal values and solution sets to stochastic programming problems is studied when the underlying probability distribution varies in some metric space of probability measures. We give conditions that imply that a stochastic program behaves stable with respect to a minimal information (m.i.) probability metric that is naturally associated with the data of the program. Canonical metrics bounding the m.i. metric are derived for specific models, namely, for linear two-stage, mixed-integer two-stage and chance constrained models. The corresponding quantivative stability results as well as some consequences for asymptotic properties of empirical approximations extend earlier results in this direction. In particular, rates of convergence in probability are derived under metric entropy conditions. Finally, we study stability properties of stable investment portfolios having minimal risk with respect to the spectral measure and stability index of the underlying stable probability distribution.
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