Irreducible Convex Paving for Decomposition of Multi-dimensional Martingale Transport Plans

Martingale transport plans on the line are known to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in higher... [ view full abstract ]

Martingale transport plans on the line are known to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in higher dimension. Our decomposition is a partition consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well-defined. We prove the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.