Integrability of solutions of the Skorokhod Embedding Problem for Diffusions

TL;DR

This paper characterizes when solutions to the Skorokhod Embedding Problem for diffusions are integrable, establishing conditions for existence, minimality, and properties of such solutions, extending classical results beyond Brownian motion.

Contribution

It provides necessary and sufficient conditions for the existence of integrable solutions to the SEP for general diffusions, and analyzes the properties of minimal solutions.

Findings

Existence of integrable solutions characterized by new conditions.

All minimal solutions share the same first moment.

For Brownian motion, all integrable embeddings are minimal.

Abstract

Suppose $X$ is a time-homogeneous diffusion on an interval $I^X \subseteq \mathbb R$ and let $\mu$ be a probability measure on $I^X$. Then $\tau$ is a solution of the Skorokhod embedding problem (SEP) for $\mu$ in $X$ if $\tau$ is a stopping time and $X_\tau \sim \mu$. There are well-known conditions which determine whether there exists a solution of the SEP for $\mu$ in $X$. We give necessary and sufficient conditions for there to exist an integrable solution. Further, if there exists a solution of the SEP then there exists a minimal solution. We show that every minimal solution of the SEP has the same first moment. When $X$ is Brownian motion, every integrable embedding of $\mu$ is minimal. However, for a general diffusion there may be integrable embeddings which are not minimal.