Instability of Martingale optimal transport in dimension d $\ge$ 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 R only. We show that the restriction to dimension d = 1 is not accidental by presenting a sequence of marginal distributions on 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 ≥ 2. For d ≥ 2 it also provides a contradiction to the martingale Wasserstein inequality established by Jourdain and Margheriti in d = 1.

Backward and Forward Wasserstein Projections in Stochastic Order

“…In view of [3, Proposition 9.3.2], P cx 2, ν is convex along generalized geodesics, hence also convex along geodesics. The forward cone P cx 2,µ , on the other hand, is generally not geodesically convex as illustrated by the example below which is inpired by [12]. a b Consider the cone P cx 2,µ .…”

“…A large amount of research has been devoted to the stability issues of optimal martingale transport. Under some conditions in one dimension, it is stable [6,26,29], but not in high dimensions [12]. These issues have become a great hindrance to the numerical pursuits of stochastic orders.…”

Computational methods for adapted optimal transport

“…Our results show that AOT is surprisingly well behaved for an OT problem with additional constraints. This is not always the case, for instance, the related martingale OT problem is not stable in dimension d ≥ 2 (at least for the weak topology on the marginals) as shown in [13], while the corresponding question in dimension d = 1 received considerable attention [7,23,37]. We further mention that computational methods for AOT problems have so far mostly been focused on backward induction, see [32,33,34].…”