An optimal transport based characterization of convex order

Abstract

For probability measures $\mu ,\nu$ and $\rho$ define the cost functionals \begin{align*} C(μ,ρ):=\sup_{π\in Π(μ,ρ)} \int \langle x,y\rangle\, π(dx,dy),\quad C(ν,ρ):=\sup_{π\in Π(ν,ρ)} \int \langle x,y\rangle\, π(dx,dy), \end{align*} where $⟨\cdot ,\cdot ⟩$ 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. $\mu {⪯}_c\nu$) 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 $\int fd\nu -\int fd\mu$ 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.