An optimal transport-based characterization of convex order

Wiesel Johannes () and Zhang Erica ()
Additional contact information
Wiesel Johannes: Department of Mathematics, Carnegie Mellon University, Wean Hall, 5000 Forbes Ave, Pittsburgh, PA 15213, USA
Zhang Erica: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA

Dependence Modeling, 2023, vol. 11, issue 1, 15

Abstract: For probability measures μ , ν \mu ,\nu , and ρ \rho , define the cost functionals C ( μ , ρ ) ≔ sup π ∈ Π ( μ , ρ ) ∫ ⟨ x , y ⟩ π ( d x , d y ) and C ( ν , ρ ) ≔ sup π ∈ Π ( ν , ρ ) ∫ ⟨ x , y ⟩ π ( d x , d y ) , C\left(\mu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\mu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y)\hspace{1.0em}{\rm{and}}\hspace{1em}C\left(\nu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\nu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y), where ⟨ ⋅ , ⋅ ⟩ \langle \cdot ,\cdot \rangle denotes the scalar product and Π ( ⋅ , ⋅ ) \Pi \left(\cdot ,\cdot ) is the set of couplings. We show that two probability measures μ \mu and ν \nu on R d {{\mathbb{R}}}^{d} with finite first moments are in convex order (i.e., μ ≼ c ν \mu {\preccurlyeq }_{c}\nu ) iff C ( μ , ρ ) ≤ C ( ν , ρ ) C\left(\mu ,\rho )\le C\left(\nu ,\rho ) holds for all probability measures ρ \rho on R d {{\mathbb{R}}}^{d} with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of ∫ f d ν − ∫ f d μ \int f{\rm{d}}\nu -\int f{\rm{d}}\mu over all 1-Lipschitz functions f f , which is obtained through optimal transport (OT) duality and the characterization result of OT (couplings) by Rüschendorf, by Rachev, and by Brenier. 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.

Keywords: convex order; optimal transport; Wasserstein distance; model-independent finance (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1515/demo-2023-0102 (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:vrs:demode:v:11:y:2023:i:1:p:15:n:1

DOI: 10.1515/demo-2023-0102

Access Statistics for this article

Dependence Modeling is currently edited by Giovanni Puccetti

More articles in Dependence Modeling from De Gruyter
Bibliographic data for series maintained by Peter Golla ().