On Skorokhod embeddings and Poisson equations

Döring, Leif; Gonon, Lukas; Prömel, David J. und Reichmann, Oleg (2019): On Skorokhod embeddings and Poisson equations. In: Annals of Applied Probability, Bd. 29, Nr. 4

Abstract

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 embed- ding 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 L ́evy processes we carry out the procedure and extend a result of Bertoin and Le Jan to L ́evy processes without local times.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Mathematik, Informatik und Statistik > Mathematik > Finanz- und Versicherungsmathematik
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik
• ISSN: 1050-5164
• Sprache: Englisch