Abstract
Weak optimal transport has been recently introduced by Gozlan et al. The original motivation stems from the theory of geometric inequalities; further applications concern numerics of martingale optimal transport and stability in mathematical finance. In this note we provide a complete geometric characterization of the weak version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain.
Highlights
There has been a growing interest in weak transport problems as introduced by Gozlan, Roberto, Samson and Tetali [14]
In this note we provide a complete geometric characterization of the weak version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain
A consequence of Theorem 1.2 is that the optimizer of (1.2) does not depend on the choice of the convex function θ. We find this fact non-trivial as well as remarkable and highlight that it is not new: different independent proofs were given by Gozlan, Roberto, Samson and Tetali [14], Alfonsi, Corbetta, Jourdain [2] and Shu [20]
Summary
There has been a growing interest in weak transport problems as introduced by Gozlan, Roberto, Samson and Tetali [14]. A consequence of Theorem 1.2 is that the optimizer of (1.2) does not depend on the choice of the convex function θ. We find this fact non-trivial as well as remarkable and highlight that it is not new: different independent proofs were given by Gozlan, Roberto, Samson and Tetali [14], Alfonsi, Corbetta, Jourdain [2] and Shu [20]. The map T can be explicitly characterized in geometric terms using the notion of irreducibility introduced in [8]: Measures η, ν ∈ P1(R) are in convex order iff their potential functions satisfy uη(y) := R |x − y|η(dx) ≤ R |x − y|ν(dx) =: uν(y). The novelty of Theorem 1.3 is that there exists only one admissible mapping with this irreducibility property
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