Finitely additive mass transportation

TL;DR

This paper extends classical mass transportation theory to finitely additive probabilities, establishing duality, attainability, and martingale transport results without additional assumptions, and provides conditions for the existence of martingale couplings.

Contribution

It introduces a finitely additive framework for mass transportation, broadening the scope of classical results and analyzing martingale transport in this setting.

Findings

Classical mass transportation results hold under finitely additive probabilities.

Existence of martingale couplings is characterized in the finitely additive setting.

Conditions for non-empty martingale transport sets are provided.

Abstract

Some classical mass transportation problems are investigated in a finitely additive setting. Let $\Omega=\prod_{i=1}^n\Omega_i$ and $\mathcal{A}=\otimes_{i=1}^n\mathcal{A}_i$, where $(\Omega_i,\mathcal{A}_i,\mu_i)$ is a ($\sigma$-additive) probability space for $i=1,\ldots,n$. Let $c:\Omega\rightarrow [0,\infty]$ be an $\mathcal{A}$-measurable cost function. Let $M$ be the collection of finitely additive probabilities on $\mathcal{A}$ with marginals $\mu_1,\ldots,\mu_n$. If couplings are meant as elements of $M$, most classical results of mass transportation theory, including duality and attainability of the Kantorovich inf, are valid without any further assumptions. Special attention is devoted to martingale transport. Let $(\Omega_i,\mathcal{A}_i)=(\mathbb{R},\mathcal{B}(\mathbb{R}))$ for all $i$ and $$M_1=\bigl\{P\in M:P\ll P^*\text{ and }(\pi_1,\ldots,\pi_n)\text{ is a }P\text{-martingale}\}$$ where $P^*$ is a reference probability on $\mathcal{A}$. If $M_1\ne\emptyset$, then $$\int c\,dP=\inf_{Q\in M_1}\int c\,dQ\quad\quad\text{for some }P\in M_1.$$ Conditions for $M_1\ne\emptyset$ are given as well.