Fine Properties of the Optimal Skorokhod Embedding Problem

Understand

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.