An optimal transport based characterization of convex order

2022/07/04 by Johannes Wiesel, Wiesel, Johannes, Erica Zhang +1 · 2 citations
Mathematics · Medicine · #Markov Chains and Monte Carlo Methods #Drug Transport and Resistance Mechanisms

paper · pdf · doi:10.48550/arxiv.2207.01235

Abstract

For probability measures μ,ν and ρ define the cost functionals C(μ,ρ):=sup_{π∈ Π(μ,ρ)} ∫ ⟨ x,y⟩ π(dx,dy), C(ν,ρ):=sup_{π∈ Π(ν,ρ)} ∫ ⟨ x,y⟩ π(dx,dy), where ⟨⋅, ⋅⟩ denotes the scalar product and Π(⋅,⋅) is the set of couplings. We show that two probability measures μ and ν on ℝ^d with finite first moments are in convex order (i.e. μ\preceq_cν) iff C(μ,ρ)≤ C(ν,ρ) holds for all probability measures ρ on ℝ^d with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of ∫ f dν-∫ f dμ 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.

Cited by

• Convex Order and Arbitrage
• Strassen's theorem for biased convex order

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