A Benamou-Brenier formulation of martingale optimal transport

TL;DR

This paper introduces a new continuous-time formulation for martingale optimal transport using a Benamou-Brenier approach, enabling analytical study of properties like existence, duality, and geodesics, with special results in one dimension.

Contribution

It presents a novel PDE-based formulation of martingale optimal transport, connecting stochastic problems with classical PDEs and analyzing properties like finiteness and geodesic structures.

Findings

Established a PDE formulation equivalent to the stochastic problem

Proved existence, duality, and geodesic properties for the new formulation

Linked geodesics to porous medium equations in one dimension

Abstract

We introduce a Benamou-Brenier formulation for the continuous-time martingale optimal transport problem as a weak length relaxation of its discrete-time counterpart. By the correspondence between classical martingale problems and Fokker-Planck equations, we obtain an equivalent PDE formulation for which basic properties such as existence, duality and geodesic equations can be analytically studied, yielding corresponding results for the stochastic formulation. In the one dimensional case, sufficient conditions for finiteness of the cost are also given and a link between geodesics and porous medium equations is partially investigated.