Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer (2019) and Wiesel (2019). We present a new perspective of this result using the theory of set-valued maps. In particular, using results from Beiglböck et al. (2021), we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals. Moreover, we establish compactness of the set of optimizers as well as upper hemicontinuity of the optimizers w.r.t. the marginals.

中文翻译：

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关于the最优运输问题的稳定性：集值映射方法

最近在Backhoff-Veraguas和Pammer（2019）和Wiesel（2019）建立了mar最优运输问题的边际边际价值的连续性。我们使用集值映射理论介绍了此结果的新观点。特别是，使用Beiglböck等人的结果。（2021），我们证明了具有固定边际的mar测度集是连续的，即上下半连续的边际。此外，我们建立了优化器集的紧致性以及优化器的上半连续性和边际性。