Martingale Benamou--Brenier: a probabilistic perspective

TL;DR

This paper introduces a martingale version of the Benamou-Brenier formulation of optimal transport, providing a probabilistic perspective that connects to classical martingales, Brownian motion, and financial models.

Contribution

It formulates a novel martingale Benamou-Brenier problem, characterizes its unique solution, and links it to classical martingales and applications in finance and PDEs.

Findings

Unique Markov-martingale solution mimics Brownian motion.

Provides a time-consistent interpolation between distributions.

Recovers classical martingales like Brownian motion and geometric Brownian motion.

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.