Non-decreasing martingale couplings

¹ CERMICS, Ecole des Ponts, INRIA, Marne-la-Vallée, France
² INRIA Paris, 48 rue Barrault, CS 61534, 75647 Paris Cedex, France; Université Paris-Dauphine; Ecole des Ponts ParisTech

^* Corresponding author: benjamin.jourdain@enpc.fr

Received: 27 November 2023
Accepted: 29 July 2024

Abstract

For many examples of couples (μ, ν) of probability measures on the real line in the convex order, we observe numerically that the Hobson and Neuberger martingale coupling, which maximizes for ρ = 1 the integral of |y − x|^ρ with respect to any martingale coupling between μ and ν, is still very close to maximize this integral for ρ ∈ (0, 2) and minimize it for ρ > 2. We investigate theoretically when it is an optimizer for ρ ≠ 1 and exhibit rather restrictive sufficient conditions. We also exhibit couples (μ, ν) such that it is not an optimizer. The support of the Hobson and Neuberger coupling is known to satisfy some monotonicity property which we call non-decreasing. We check that the non-decreasing property is preserved for maximizers when ρ ∈ (0, 1]. In general, there exist distinct non-decreasing martingale couplings, and we find some decomposition of ν which is in one-to-one correspondence with martingale couplings non-decreasing in a generalized sense.