On Skorokhod Embeddings and Poisson Equations

Leif Döring, Lukas Gonon, David J. Prömel, Oleg Reichmann

Published 2017-03-16Version 1

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 and the generator $\mathcal A$ of $X$ has a formal adjoint operator, then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation for the adjoint operator and give an explicit construction of the stopping time using the solution of the Poisson equation. For the class of L\'evy processes we carry out the procedure and show how to generalize the classical convex order condition for the Brownian motion. In particular, our technique extends a result of Bertoin and Le Jan to L\'evy processes without local times.