On Skorokhod embeddings and Poisson equations

The classical Skorokhod embedding problem for a Brownian motion W asks to find a stopping time τ so that Wτ is distributed according to a prescribed probability distribution μ. 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 μ0 and let μ1 be a probability measure. The task is to find a stopping time τ such that Xτ is distributed according to μ1. More precisely, we study the question of deciding if a finite mean solution to the SEP can exist for given μ0,μ1 and the task of giving a solution which is as explicit as possible. If μ0 and μ1 have positive densities h0 and h1 and the generator A of X has a formal adjoint operator A∗, then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation A∗H=h1−h0 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