Martingale Benamou.Brenier: A probabilistic perspective

In classical optimal transport, the contributions of Benamou.Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou.Brenier type formulation of the martingale transport problem for given d -dimensional distributions ., . in convex order. The unique solution M.=(M.t)t.[0,1] of this problem turns out to be a Markov-martingale which has several notable properties: In a specific sense it mimics the movement of a Brownian particle as closely as possible subject to the conditions M.0.., M.1... Similar to McCann.s displacement-interpolation, M. provides a time-consistent interpolation between . and .. For particular choices of the initial and terminal law, M. recovers archetypical martingales such as Brownian motion, geometric Brownian motion, and the Bass martingale. Furthermore, it yields a natural approximation to the local vol model and a new approach to Kellerer.s theorem. This article is parallel to the work of Huesmann.Trevisan, who consider a related class of problems from a PDE-oriented perspective.