An optimal transport based characterization of convex order

TL;DR

This paper characterizes convex order between probability measures using optimal transport cost functionals, providing new theoretical insights, proofs, computational methods, and applications in finance.

Contribution

It generalizes a known characterization of convex order via optimal transport costs to higher dimensions and introduces new proofs, computational techniques, and financial applications.

Findings

Convex order characterized by cost inequalities for all bounded support measures.

New proofs of one-dimensional convex order characterizations.

Applications to model-independent arbitrage strategies in finance.

Abstract

For probability measures $\mu,\nu$ and $\rho$ define the cost functionals \begin{align*} C(\mu,\rho):=\sup_{\pi\in \Pi(\mu,\rho)} \int \langle x,y\rangle\, \pi(dx,dy),\quad C(\nu,\rho):=\sup_{\pi\in \Pi(\nu,\rho)} \int \langle x,y\rangle\, \pi(dx,dy), \end{align*} where $\langle\cdot, \cdot\rangle$ denotes the scalar product and $\Pi(\cdot,\cdot)$ is the set of couplings. We show that two probability measures $\mu$ and $\nu$ on $\mathbb{R}^d$ with finite first moments are in convex order (i.e. $\mu\preceq_c\nu$) iff $C(\mu,\rho)\le C(\nu,\rho)$ holds for all probability measures $\rho$ on $\mathbb{R}^d$ with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of $\int f\,d\nu -\int f\,d\mu$ over all $1$-Lipschitz functions $f$, which is obtained through optimal transport duality and Brenier's theorem. Building on this result, we derive new proofs of well-known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.