A Robust Version of Convex Integral Functionals

K. Owari

Journal of convex analysis, Tome 22 (2015) no. 3, pp. 827-852

Résumé

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.

Détail

Classification : 46N10, 46E30, 49N15, 52A41, 91G80
Mots-clés : Convex integral functionals, duality, robust stochastic optimization, financial mathematics