Abstract

We study the pointwise supremum of convex integral functionals If,γ(ξ)=sup⁡Q(∫Ωf(ω,ξ(ω))Q(dω)−γ(Q))\mathcal{I}_{f,\gamma}(\xi)= \sup_{Q} \left( \int_\Omega f(\omega,\xi(\omega))Q(d\omega)-\gamma(Q)\right) on L∞(Ω,F,P)L^\infty(\Omega,\mathcal{F},\mathbb{P}) where f:Ω×R→R‾f:\Omega \times\mathbb{R}\rightarrow\overline{\mathbb{R}} is a proper normal convex integrand, γ\gamma is a proper convex function on the set of probability measures absolutely continuous w.r.t. P\mathbb{P}, and the supremum is taken over all such measures. We give a pair of upper and lower bounds for the conjugate of If,γ\mathcal{I}_{f,\gamma} as direct sums of a common regular part and respective singular parts; they coincide when dom(γ)={P}\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.

Author information

Contact details are reproduced from the original publication and may be historical.

Keita Owari

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

owari@e.u-tokyo.ac.jp

Keywords

  • Convex integral functionals
  • duality
  • robust stochastic optimization
  • financial mathematics

Mathematics Subject Classification

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

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Suggested citation

K. Owari. “A Robust Version of Convex Integral Functionals.” Journal of Convex Analysis 22 (2015), No. 3, 827–852. https://doi.org/10.68381/jca22042

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.