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On the stability of the martingale optimal transport problem: A set-valued map approach

2021; Elsevier BV; Volume: 176; Linguagem: Inglês

10.1016/j.spl.2021.109131

1879-2103

Ariel Neufeld, Julian Sester,

Markov Chains and Monte Carlo Methods

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.