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Journal of Convex Analysis 09 (2002), No. 1, 237--244
Copyright Heldermann Verlag 2002

Convex Stochastic Duality and the "Biting Lemma"

Igor V. Evstigneev
School of Economic Studies, University of Manchester, Oxford Road, Manchester M13 9PL, Great Britain

Sjur D. Flåm
Dept. of Economics, University of Bergen, Fosswickels gate 6, 5007 Bergen, Norway


A standard approach to duality in stochastic optimization problems with constraints in $L_{\infty}$ relies upon the Yosida - Hewitt theorem. We develop an alternative technique which employs only "elementary" means. The technique is based on an $\varepsilon$-regularization of the original problem and on passing to the limit as $\varepsilon \to 0$ with the help of a simple measure-theoretic fact -- the biting lemma.

Keywords: Stochastic optimization, convex duality, constraints in L-infinity, stochastic Lagrange multipliers, bounded sets in L-1, biting lemma, Gale's economic model.

MSC: 90C15, 51A41; 90C19, 90A16

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