On the monotonicity principle of optimal Skorokhod embedding problem

TL;DR

This paper offers a new proof of the monotonicity principle for the optimal Skorokhod embedding problem, using advanced duality and measure-theoretic techniques to clarify the geometric nature of optimal embeddings.

Contribution

It provides an alternative proof of the monotonicity principle, enhancing understanding of the geometric and optimality properties in the Skorokhod embedding problem.

Findings

New proof of the monotonicity principle using duality methods

Clarification of the geometric characterization of optimal embeddings

Application of measure-theoretic tools to the problem

Abstract

In this paper, we provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established by Beiglb\"ock, Cox and Huesmann. This principle presents a geometric characterization that reflects the desired optimality properties of Skorokhod embeddings. Our proof is based on the adaptation of the Monge-Kantorovich duality in our context together with a delicate application of the optional cross-section theorem and a clever conditioning argument.