On Skorokhod embeddings and Poisson equations

Abstract

The classical Skorokhod embedding problem for a Brownian motion $W$ asks to find a stopping time $\tau $ so that $W_{\tau }$ is distributed according to a prescribed probability distribution $\mu $. Many solutions have been proposed during the past 50 years and applications in different fields emerged. This article deals with a generalized Skorokhod embedding problem (SEP): Let $X$ be a Markov process with initial marginal distribution $\mu_{0}$ and let $\mu_{1}$ be a probability measure. The task is to find a stopping time $\tau $ such that $X_{\tau }$ is distributed according to $\mu_{1}$. More precisely, we study the question of deciding if a finite mean solution to the SEP can exist for given $\mu_{0}$, $\mu_{1}$ and the task of giving a solution which is as explicit as possible. If $\mu_{0}$ and $\mu_{1}$ have positive densities $h_{0}$ and $h_{1}$ and the generator $\mathcal{A}$ of $X$ has a formal adjoint operator $\mathcal{A}^{*}$, then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation $\mathcal{A}^{*}H=h_{1}-h_{0}$ and give a fairly explicit construction of the stopping time using the solution of the Poisson equation. For the class of Lévy processes, we carry out the procedure and extend a result of Bertoin and Le Jan to Lévy processes without local times.