Electronic Journal of Probability

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

Alexander Cox and Jan Obloj

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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.

Article information

Electron. J. Probab., Volume 13 (2008), paper no. 42, 1203-1228.

Accepted: 31 July 2008
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 28A80: Fractals [See also 37Fxx]

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

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Cox, Alexander; Obloj, Jan. Classes of measures which can be embedded in the Simple Symmetric Random Walk. Electron. J. Probab. 13 (2008), paper no. 42, 1203--1228. doi:10.1214/EJP.v13-516. https://projecteuclid.org/euclid.ejp/1464819115

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