Duality for increasing convex functionals with countably many marginal constraints

TL;DR

This paper develops a duality framework for increasing convex functionals with countably many marginal constraints, extending classical transport dualities to more general settings involving expectations and maxima.

Contribution

It introduces a convex dual representation for such functionals, generalizing Kantorovich and martingale transport dualities to countably infinite product spaces.

Findings

Derived a convex dual representation under marginal tightness conditions

Extended Kantorovich's duality to countably many marginals

Connected martingale transport duality with convex functional duality

Abstract

In this work we derive a convex dual representation for increasing convex functionals on a space of real-valued Borel measurable functions defined on a countable product of metric spaces. Our main assumption is that the functionals fulfill marginal constraints satisfying a certain tightness condition. In the special case where the marginal constraints are given by expectations or maxima of expectations, we obtain linear and sublinear versions of Kantorovich's transport duality and the recently discovered martingale transport duality on products of countably many metric spaces.