Finitely additive mass transportation

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.