Integrability of solutions of the Skorokhod Embedding Problem for Diffusions

Suppose $X$ is a time-homogeneous diffusion on an interval $I^X \subseteq \mathbb R$ and let $μ$ be a probability measure on $I^X$. Then $τ$ is a solution of the Skorokhod embedding problem (SEP) for $μ$ in $X$ if $τ$ is a stopping time and $X_τ\sim μ$. There are well-known conditions which determine whether there exists a solution of the SEP for $μ$ 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 $μ$ is minimal. However, for a general diffusion there may be integrable embeddings which are not minimal.

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