MARTINGALE BENAMOU–BRENIER: A PROBABILISTIC PERSPECTIVE

Abstract

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 μ, ν in convex order. The unique solution M* = (M* t )t∊[0,1] of this problem turns out to be a Markovmartingale 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 ~μ, M*1m ~ν. Similar to McCann’s displacement-interpolation, M* provides a time-consistent interpolation between μ and ν. 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.