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Discrete Higher Order Convex Functions and their Applications

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Generalized Convexity and Generalized Monotonicity

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 502))

Abstract

In this paper we present an overview about the recently developed theory of discrete moment problems, i.e., moment problems where the supports of the random variables involved are discrete. We look for the minimum or maximum of a linear functional acting on an unknown probability distribution subject to a finite number of moment constraints. Using linear programming methodology, we present structural theorems, in both the univariate and multivariate cases, for the dual feasible bases and show how the relevant problems can be solved by suitable adaptations of the dual method. The condition on the objective function is a kind of higher order convexity, expressed in terms of divided differences. A variant of the above, the discrete binomial moment problem, as well as generalization for discrete variable Chebyshev systems are also discussed. Finally, we present novel applications to valuations of financial instruments.

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Author information

Authors and Affiliations

  1. RUTCOR, Rutgers Center for Operations Research, 640 Bartholomew Road, Piscataway, NJ, 08854-8003, USA

    András Prékopa

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  1. András Prékopa
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of the Aegean, 83200, Karlovassi/Samos, Greece

    Nicolas Hadjisavvas

  2. CODE and Department of Economics, Universitat Autonoma de Barcelona, 08193, Bellaterra, Spain

    Juan Enrique Martínez-Legaz

  3. Department of Mathematics Faculty of Sciences, Université de Pau, 64000, Pau, France

    Jean-Paul Penot

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© 2001 Springer-Verlag Berlin Heidelberg

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Prékopa, A. (2001). Discrete Higher Order Convex Functions and their Applications. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, JP. (eds) Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems, vol 502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56645-5_2

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  • DOI: https://doi.org/10.1007/978-3-642-56645-5_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41806-1

  • Online ISBN: 978-3-642-56645-5

  • eBook Packages: Springer Book Archive

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