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Classes of measures which can be embedded in the Simple Symmetric Random Walk

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Classes of measures which can be embedded in the Simple Symmetric Random Walk
2.	Creator	Author's name, affiliation, country	Alexander M.G. Cox; University of Bath
2.	Creator	Author's name, affiliation, country	Jan K. Obloj; Imperial College London
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Skorokhod embedding problem; random walk; minimal stopping time; Azema-Yor stopping time; Chacon-Walsh stopping time; iterated function system; self-similar set; fractal; uniform integrability
3.	Subject	Subject classification	60G50, 60G40, 28A80
4.	Description	Abstract	We characterize the possible distributions of a stopped simple symmetric random walk $X_\tau$, where $\tau$ is a stopping time relative to the natural filtration of $(X_n)$. We prove that any probability measure on $\mathbb{Z}$ can be achieved as the law of $X_\tau$ where $\tau$ is a minimal stopping time, but the set of measures obtained under the further assumption that $(X_{n\land \tau}:n\geq 0)$ is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on $\mathbb{Z}$. This is in sharp contrast to the well-studied Brownian motion setting. We also investigate the discrete counterparts of the Chacon-Walsh (1976) and Azema-Yor (1979) embeddings and show that they lead to yet smaller sets of achievable measures. Finally, we solve explicitly the Skorokhod embedding problem constructing, for a given measure $\mu$, a minimal stopping time $\tau$ which embeds $\mu$ and which further is uniformly integrable whenever a uniformly integrable embedding of $\mu$ exists.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Nuffield Foundation; 6th European Community Framework Programme
7.	Date	(YYYY-MM-DD)	2008-07-31
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/516
10.	Identifier	Digital Object Identifier	10.1214/EJP.v13-516
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 13
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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