Papers › An optimal transport based characterization of convex order
An optimal transport based characterization of convex order
Johannes Wiesel, Erica Zhang
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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 ℝᵈ with finite first moments are in convex order (i.e. μ≼_cν) iff C(μ,ρ)≤C(ν,ρ) holds for all probability measures ρ on ℝᵈ 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.
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