A construction of the left-curtain coupling

In a martingale optimal transport (MOT) problem mass distributed according to the law $\mu$ is transported to the law $\nu$ in such a way that the martingale property is respected. Beiglb\"ock and Juillet (On a problem of optimal transport under marginal martingale constraints, Annals of Probability, 44(1):42-106, 2016) introduced a solution to the MOT problem which they baptised the left-curtain coupling. The left-curtain coupling has been widely studied and shown to have many applications, including to martingale inequalities and the model-independent pricing of American options. Beiglb\"ock and Juillet proved existence and uniqueness, proved optimality for a family of cost functions, and proved that when $\mu$ is a continuous distribution, mass at $x$ is mapped to one of at most two points, giving lower and upper functions. Henry-Labord\`ere and Touzi (An explicit martingale version of Brenier`s theorem, Finance and Stochastics, 20:635-668, 2016) showed that the left-curtain coupling is optimal for an extended family of cost functions and gave a construction of the upper and lower functions under an assumption that $\mu$ and $\nu$ are continuous, together with further simplifying assumptions of a technical nature. In this article we construct these upper and lower functions in the general case of arbitrary centred measures in convex order, and thereby give a complete construction of the left-curtain coupling. In the case where $\mu$ has atoms these upper and lower functions are to be interpreted in the sense of a lifted martingale.