{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-characterisation-of-convex-order-using-the","title":"An optimal transport based characterization of convex order","arxiv_id":"2207.01235","date":"2022-07-04","proceeding":null,"authors":["Johannes Wiesel","Erica Zhang"],"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.","url_abs":"https://arxiv.org/abs/2207.01235v3","url_pdf":"https://arxiv.org/pdf/2207.01235v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-characterisation-of-convex-order-using-the","repo_url":"https://github.com/johanneswiesel/convex-order","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}