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


Cox, A. M. and Obloj, J., 2008. Classes of measures which can be embedded in the Simple Symmetric Random Walk. Electronic Journal of Probability, 13, pp. 1203-1228.

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    We characterize the possible distributions of a stopped simple symmetric random walk Xτ, where τ is a stopping time relative to the natural filtration of (Xn). We prove that any probability measure on Z can be achieved as the law of X stopped at a minimal stopping time, but the set of measures obtained under the further assumption that stopped process is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on 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 Azéma-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 μ, a minimal stopping time τ which embeds μ and which further is uniformly integrable whenever a uniformly integrable embedding of μ exists.


    Item Type Articles
    CreatorsCox, A. M.and Obloj, J.
    DepartmentsFaculty of Science > Mathematical Sciences
    ID Code27002


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