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Journal of Convex Analysis 22 (2015), No. 3, 827--852
Copyright Heldermann Verlag 2015

A Robust Version of Convex Integral Functionals

Keita Owari
Graduate School of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan


We study the pointwise supremum of convex integral functionals $$ \mathcal{I}_{f,\gamma}(\xi)= \linebreak\sup_{Q} \left( \int_\Omega f(\omega,\xi(\omega))Q(d\omega)-\gamma(Q)\right) $$ on $L^\infty(\Omega,\mathcal{F},\mathbb{P})$ where $f:\Omega \times\mathbb{R}\rightarrow\overline{\mathbb{R}}$ is a proper normal convex integrand, $\gamma$ is a proper convex function on the set of probability measures absolutely continuous w.r.t. $\mathbb{P}$, and the supremum is taken over all such measures. We give a pair of upper and lower bounds for the conjugate of $\mathcal{I}_{f,\gamma}$ as direct sums of a common regular part and respective singular parts; they coincide when $\mathrm{dom}(\gamma)=\{\mathbb{P}\}$ as Rockafellar's classical result, while both inequalities can generally be strict. We then investigate when the conjugate eliminates the singular measures, which a fortiori yields the equality in bounds, and its relation to other finer regularity properties of the original functional and of the conjugate.

Keywords: Convex integral functionals, duality, robust stochastic optimization, financial mathematics.

MSC: 46N10, 46E30, 49N15, 52A41, 91G80

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