Bi-martingale optimal transport and its applications

TL;DR

This paper introduces a novel bi-martingale optimal transport framework that connects various variational problems on probability measures, providing new interpretations and robust approximation schemes for classical martingale optimal transport, especially in higher dimensions.

Contribution

It develops a new non-linear optimal transport formulation with martingale constraints, linking it to Zolotarev distances and convex order problems, and offers a stable approximation method for martingale optimal transport.

Findings

Provides an optimal transport interpretation of the second Zolotarev distance.

Develops a bi-martingale approximation scheme for classical martingale optimal transport.

Addresses instability issues of MOT with respect to marginal variations in higher dimensions.

Abstract

We introduce a new non-linear optimal transport formulation for a pair of probability measures on $\mathbb{R}^d$ sharing a common barycentre, in which admissible transference plans satisfy two martingale-type constraints. This bi-martingale framework underlies and interconnects several variational problems on the space of probability measures. For the quadratic cost, it provides an optimal transport interpretation of the second Zolotarev distance on $\mathrm{P}_2(\mathbb{R}^d)$. For a broader class of convex costs, it leads to optimization problems under convex order constraints, encompassing in particular the Zolotarev projection onto the cone of dominating probability measures. As a main application, we construct a $\Gamma$-convergent bi-martingale approximation of the classical martingale optimal transport problem. This scheme robustly accommodates deviations from convex order between the marginal distributions and overcomes the well-known instability of MOT with respect to variations of the marginals in higher dimensions.