A Benamou Brenier approach to martingale optimal transport

We introduce and analyze a continuous time martingale optimal transport problem (MOT) which can be seen as the "Benamou-Brenier" formulation of MOT. It is naturally linked to the discrete MOT problem via a weak length relaxation. We present two different solutions to this problem. The first solution is based on a convex duality result and allows to derive a "geodesic equation" for the optimizer for a wide class of cost functions. The second is an explicit probabilistic representation in the case of a specific cost function. We will show that this solution has several applications as well as a remarkable additional optimality property.

(based on joint work with Julio Backhoff, Mathias Beiglböck, Sigrid Källblad, and Dario Trevisan)