Döring, Leif; Gonon, Lukas; Prömel, David J.; Reichmann, Oleg On Skorokhod embeddings and Poisson equations. (English) Zbl 1466.60083 Ann. Appl. Probab. 29, No. 4, 2302-2337 (2019). Summary: 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 \(\mathcal{A}^\ast\), then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation \(\mathcal{A}^\ast H=h_1-h_0\) 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 J. Bertoin and Y. Le Jan [Ann. Probab. 20, No. 1, 538–548 (1992; Zbl 0749.60038)] to Lévy processes without local times. 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