Continuity of the martingale optimal transport problem on the real line

Abstract

We show continuity of the martingale optimal transport optimisation problem as a functional of its marginals. This is achieved via an estimate on the projection in the nested/causal Wasserstein distance of an arbitrary coupling on to the set of martingale couplings with the same marginals. As a corollary we obtain an independent proof of sufficiency of the monotonicity principle established in Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) for cost functions of polynomial growth.

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