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Journal of Convex Analysis 24 (2017), No. 4, 1375--1405
Copyright Heldermann Verlag 2017

Representation of Convex Operators and their Static and Dynamic Sandwich Extensions

Jocelyne Bion-Nadal
Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau, France

Giulia Di Nunno
Dept. of Mathematics, University of Oslo, P. O. Box 1053 Blindern, 0316 Oslo, Norway

Monotone convex operators and time-consistent systems of operators appear naturally in stochastic optimisation and mathematical finance in the context of pricing and risk measurement. We study the dual representation of a monotone convex operator when its domain is defined on a subspace of Lp, with 1 ≤ p ≤ ∞, and we prove a sandwich preserving extension theorem. These results are then applied to study systems of such operators defined only on subspaces. We propose various dynamic sandwich preserving extension results depending on the nature of time: finite discrete, countable discrete, and continuous. Of particular notice is the fact that the extensions obtained are time-consistent.

Keywords: Convex operator, sandwich preserving extension, dual representation, time consistency, dynamic risk measures, price operator.

MSC: 46A20, 47N30, 91B25, 91B70, 52A41.

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