An optimal transport-based characterization of convex order

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

Publication date: 24 January 2024

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

Abstract: For probability measures

m u, u

and

h o

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

l a n g l e c d o t, c d o t a n g l e

denotes the scalar product and

P i (c d o t, c d o t)

is the set of couplings. We show that two probability measures

m u

and

u

on

m a t h b b R^{d}

with finite first moments are in convex order (i.e.

m u p r e c e q_{c} u

) iff

C (m u, h o) l e C (u, h o)

holds for all probability measures

h o

on

m a t h b b R^{d}

with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of

i n t f, d u - i n t f, d m u

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 distance optimal transport convex order model-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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