Quasi-sure duality for multi-dimensional martingale optimal transport

We provide a multi-dimensional quasi sure duality for the martingale optimal transport problem. We also prove a disintegration result which states a natural decomposition of the martingale optimal transport problem on the... [ view full abstract ]

We provide a multi-dimensional quasi sure duality for the martingale optimal transport problem. We also prove a disintegration result which states a natural decomposition of the martingale optimal transport problem on the irreducible components, with pointwise duality verified on each component. We also extend the martingale monotonicity principle to the present multi-dimensional setting. Our results hold in dimensions 1, 2, in dimension 3 provided that the target measure is dominated by Lebesgue, or under the Continuum Hypothesis. We finally provide a counterexample showing that smoothness conditions on the coupling function do not guarantee pointwise duality in dimension higher than 2.