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 only. We show that the restriction to dimension is not accidental by presenting a sequence of marginal distributions on 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 . For it also provides a contradiction to the martingale Wasserstein inequality established by Jourdain and Margheriti in .
MB is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics–Geometry–Structure.
We thank the anonymous referees for interesting comments and the idea to discuss the impact of irreducible components in Remark 3.2 (iii). Another suggestion of the referees is that an additional condition may restore the stability in dimension . We are looking forward for such an evolution in the topic.
"Instability of martingale optimal transport in dimension ." Electron. Commun. Probab. 27 1 - 10, 2022. https://doi.org/10.1214/22-ECP463