An optimal transport-based characterization of convex order

DOI10.1515/DEMO-2023-0102zbMATH Open1528.60036arXiv2207.01235OpenAlexW4387757271MaRDI QIDQ6184350

Johannes Wiesel, Erica Zhang

Publication date: 24 January 2024

Published in: Dependence Modeling (Search for Journal in Brave)

Abstract: For probability measures $mu,u$ and $ho$ define the cost functionals �egin{align*} C(mu, ho):=sup_{piin Pi(mu, ho)} int langle x,y angle, pi(dx,dy),quad C(

u, ho):=sup_{piin Pi(

u, ho)} int langle x,y angle, pi(dx,dy), end{align*} where $langlecdot,cdotangle$ denotes the scalar product and $Pi(cdot,cdot)$ is the set of couplings. We show that two probability measures $mu$ and $u$ on $mathbb{R}^d$ with finite first moments are in convex order (i.e. $muprece{q}_{c}u$) iff $C(mu,ho)leC(u,ho)$ holds for all probability measures $ho$ on $mathbb{R}^d$ with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of $intf,du-intf,dmu$ 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.

Full work available at URL: https://arxiv.org/abs/2207.01235

zbMATH Keywords

Wasserstein distanceoptimal transportconvex ordermodel-independent finance

Mathematics Subject Classification ID

Inequalities; stochastic orderings (60E15) Martingales with discrete parameter (60G42) Financial applications of other theories (91G80) Martingales and classical analysis (60G46)

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