Fine properties of the optimal Skorokhod embedding problem

Abstract

We study the problem of stopping a Brownian motion at a given distribution $\nu$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $\mathcal{T}(\nu )$ of stopping times embedding $\nu$ is weakly dense in the set $\mathcal{R}(\nu )$ of randomized embeddings. In particular, the optimal Skorokhod embedding problem over $\mathcal{T}(\nu )$ has the same value as the relaxed one over $\mathcal{R}(\nu )$ when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kantorovich couplings in optimal transport. A second part studies the dual optimization in the sense of linear programming. While existence of a dual solution failed in previous formulations, we introduce a relaxation of the dual problem that exploits a novel compactness property and yields existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of Beiglböck, Cox and Huesmann [Optimal transport and Skorokhod embedding, Invent. Math. 208, 327–400 (2017)]. We show that these results can be applied to characterize the geometry of optimal embeddings through a variational condition.