Optimal skorokhod embedding under finitely many marginal constraints

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem in Beiglböck, Cox, and Huesmann [Optimal Transport and Skorokhod Embedding, Preprint, 2013] to the case of finitely many marginal constraints. Using the classical convex duality approach together with the optimal stopping theory, we establish some duality results under more general conditions than Beiglböck, Cox, and Huesmann. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints.