Stability of the shadow projection and the left-curtain coupling

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The (left-)curtain coupling, introduced by Beiglböck and the author is an extreme element of the set of "martingale" couplings between two real probability measures in convex order. It enjoys remarkable properties with respect to order relations and a minimisation problem inspired by the theory of optimal transport. An explicit representation and a number of further noteworthy attributes have recently been established by Henry-Labordère and Touzi. In the present paper we prove that the curtain coupling depends continuously on the prescribed marginals and quantify this with Lipschitz estimates. Moreover, we investigate the Markov composition of curtain couplings as a way of associating Markovian martingales with peacocks.