Optimal Transport and Skorokhod Embedding

TL;DR

This paper introduces a novel geometric approach to the Skorokhod embedding problem using optimal mass transport, unifying existing solutions and enabling the construction of new embeddings for various Markov processes.

Contribution

It develops a systematic, geometric method for constructing optimal Skorokhod embeddings, unifying previous solutions and extending applicability to all regular Markov processes.

Findings

Unified geometric characterization of Skorokhod embeddings

Derivation of all known optimal embeddings as special cases

Introduction of new embeddings based on the framework

Abstract

The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.