@article{EJP516,
author = {Alexander Cox and Jan Obloj},
title = {Classes of measures which can be embedded in the Simple Symmetric Random Walk},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {13},
year = {2008},
keywords = {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},
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.},
pages = {no. 42, 1203-1228},
issn = {1083-6489},
doi = {10.1214/EJP.v13-516},
url = {http://ejp.ejpecp.org/article/view/516}}