Instability of martingale optimal transport in dimension d≥2

Abstract

Stability of the value function and the set of minimizers w.r.t. the given data is a desirable feature of optimal transport problems. For the classical Kantorovich transport problem, stability is satisfied under mild assumptions and in general frameworks such as the one of Polish spaces. However, for the martingale transport problem several works based on different strategies established stability results for $\mathbb{R}$only. We show that the restriction to dimension $d=1$ is not accidental by presenting a sequence of marginal distributions on ${\mathbb{R}}^2$ for which the martingale optimal transport problem is neither stable w.r.t. the value nor the set of minimizers. Our construction adapts to any dimension $d\ge 2$. For $d\ge 2$ it also provides a contradiction to the martingale Wasserstein inequality established by Jourdain and Margheriti in $d=1$.