Stability of the shadow projection and the left-curtain coupling

TL;DR

This paper studies the stability and continuity properties of the left-curtain coupling in martingale optimal transport, providing Lipschitz estimates and exploring its Markovian composition for peacocks.

Contribution

It proves the continuous dependence of the curtain coupling on marginals and analyzes its Markov composition, advancing understanding of martingale couplings.

Findings

Curtain coupling depends continuously on marginals with Lipschitz estimates.

Markov composition of curtain couplings can generate Markovian martingales.

Provides explicit bounds and stability results for the coupling.

Abstract

The (left-)curtain coupling, introduced by Beiglb\"ock 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-Labord\`ere 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.