Integrability of solutions of the Skorokhod embedding problem for diffusions

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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 ofthe 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, there exists an integrable embedding of $\mu$ if and only if $\mu$ is centred and in $L^2$. Further,every integrable embedding is minimal. When $X$ is a general time-homogeneous diffusion the situation is more subtle. The case with drift can be reduced to the local martingale case by a change of scale. If $Y$ is a diffusion in natural scale, and if the target law is centred, then as in the Brownian case, there is an integrable embedding if the target law satisfies an integral condition. However, unlike in the Brownian case, there exist integrable embeddings of target laws which are not centred. Further, there exist integrable embeddings which are not minimal. Instead, if there exists an integrable embedding, then the set of minimal embeddings is the set of embeddings such that the mean equals a certain quantity, which we identify.