The Annals of Applied Probability

On Skorokhod embeddings and Poisson equations

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

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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.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 4 (2019), 2302-2337.

Dates
Received: December 2017
Revised: July 2018
First available in Project Euclid: 23 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1563869044

Digital Object Identifier
doi:10.1214/18-AAP1454

Mathematical Reviews number (MathSciNet)
MR3984255

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J75: Jump processes

Keywords
Fokker–Planck equation Lévy process Markov process Skorokhod embedding problem random time-change

Citation

Döring, Leif; Gonon, Lukas; Prömel, David J.; Reichmann, Oleg. On Skorokhod embeddings and Poisson equations. Ann. Appl. Probab. 29 (2019), no. 4, 2302--2337. doi:10.1214/18-AAP1454. https://projecteuclid.org/euclid.aoap/1563869044


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