An extended footnote on finitely minimal martingale measures

TL;DR

This paper discusses conditions under which finitely minimal martingale measures solve the martingale transport problem, providing a proof of uniqueness for left-monotone plans under certain cost function conditions.

Contribution

It establishes that finitely minimal martingale measures are solutions under specific cost function assumptions and offers a clear proof of the uniqueness of left-monotone plans.

Findings

Finitely minimal martingale measures solve the martingale transport problem under certain conditions.

A transparent proof of the uniqueness of left-monotone martingale transport plans is provided.

Conditions on the cost function include upper semi-continuity and boundedness by integrable functions.

Abstract

This note contains a short discussion on the sufficiency of finite optimality in martingale transport. It is shown that finitely minimal martingale measures are solutions of the martingale transport problem when the cost function is upper semi-continuous and bounded from above by a sum of integrable functions. As an application a transparent proof of the uniqueness of left-monotone martingale transport plans is given.