Finitely additive mass transportation

Abstract

Some classical mass transportation problems are investigated in a finitely additive setting. Let $\mathrm{\Omega }={\prod }_{i=1}^n{\mathrm{\Omega }}_i$ and $\mathcal{A}={\otimes }_{i=1}^n{\mathcal{A}}_i$, where $({\mathrm{\Omega }}_i,{\mathcal{A}}_i,{\mathrm{\mu }}_i)$ is a (σ-additive) probability space for $i=1,\dots ,n$. Let $c:\mathrm{\Omega }\to [0,\mathrm{\infty }]$ be an $\mathcal{A}$-measurable cost function. Let M be the collection of finitely additive probabilities on $\mathcal{A}$ with marginals ${\mathrm{\mu }}_1,\dots ,{\mathrm{\mu }}_n$. 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 $({\mathrm{\Omega }}_i,{\mathcal{A}}_i)=(\mathbb{R},\mathcal{B}(\mathbb{R}))$ for all i and

$$M_1=\{P\in M:P\ll P^{\ast }\text{and}({\mathrm{\pi }}_1,\dots ,{\mathrm{\pi }}_n)\text{is a}P\text{-martingale}\}$$

where $P^{\ast }$ is a reference probability on $\mathcal{A}$ and ${\mathrm{\pi }}_1,\dots ,{\mathrm{\pi }}_n$ are the canonical projections on $\mathrm{\Omega }={\mathbb{R}}^n$. If $M_1\ne \varnothing$, the Kantorovich inf over $M_1$ is attained, in the sense that $\int cdP={inf}_{Q\in M_1}\int cdQ$ for some $P\in M_1$. Conditions for $M_1\ne \varnothing$ are given as well.