ON SKOROKHOD EMBEDDINGS AND POISSON EQUATIONS

• 成果类型: Article
• 署名作者: Doering, Leif; Gonon, Lukas; Proemel, David J.; Reichmann, Oleg
• 署名单位: University of Mannheim; University of Oxford
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1454
• 发表日期: 2019
• 页码: 2302-2337
• 关键词: time-homogeneous diffusions; stopping distributions

摘要

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 A of X has a formal adjoint operator A*, then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation 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 Levy processes, we carry out the procedure and extend a result of Bertoin and Le Jan to Levy processes without local times.