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 $\mu $, $\nu $ in convex order. The unique solution $M^{*}=(M_{t}^{*})_{t\in [0,1]}$ 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}\sim \mu $, $M^{*}_{1}\sim \nu $. Similar to McCann’s displacement-interpolation, $M^{*}$ provides a time-consistent interpolation between $\mu $ and $\nu $. 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.