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 ${\mathcal{A}}^{\ast }$, 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 Bertoin and Le Jan to Lévy processes without local times.

Journal version available at http://projecteuclid.org/euclid.aoap/1563869044 Second part of the original manuscript can be found in arXiv:1812.08579

References in corpus (4)

• The Skorokhod embedding problem and its offspring
• Gambling in contests with random initial law
• An explicit Skorokhod embedding for spectrally negative Levy processes
• Existence and uniqueness results for time-inhomogeneous time-change equations and Fokker--Planck equations

Cited by in corpus (2)

• A Free Boundary Characterisation of the Root Barrier for Markov Processes
• Existence and uniqueness results for time-inhomogeneous time-change equations and Fokker--Planck equations