Measurability of optimal transportation and strong coupling of martingale measures

TL;DR

This paper studies the measurability of optimal transportation maps with parameterized marginals and applies these results to construct strong couplings of martingale measures, providing quantitative Wasserstein distance estimates.

Contribution

It establishes joint measurability of optimal transport maps with respect to space and parameters, enabling strong couplings of martingale measures with explicit Wasserstein bounds.

Findings

Proved joint measurability of optimal transport maps.

Constructed strong couplings of martingale measures.

Provided quantitative Wasserstein distance estimates.

Abstract

We consider the optimal mass transportation problem in $\RR^d$ with measurably parameterized marginals, for general cost functions and under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with specified covariance measure. This is done by pushing forward one martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures.