Probability Theory and Related Fields

, Volume 165, Issue 1–2, pp 483–508 | Cite as

Skorokhod embeddings for two-sided Markov chains

Article

Abstract

Let \((X_n :n\in \mathbb {Z})\) be a two-sided recurrent Markov chain with fixed initial state \(X_0\) and let \(\nu \) be a probability measure on its state space. We give a necessary and sufficient criterion for the existence of a non-randomized time T such that \((X_{T+n} :n\in \mathbb {Z})\) has the law of the same Markov chain with initial distribution \(\nu \). In the case when our criterion is satisfied we give an explicit solution, which is also a stopping time, and study its moment properties. We show that this solution minimizes the expectation of \(\psi (T)\) in the class of all non-negative solutions, simultaneously for all non-negative concave functions \(\psi \).

Keywords

Skorokhod embedding Stopping time Markov chain Random walk Extra head scheme Unbiased shift Random allocation Stable matching Optimal transport 

Mathematics Subject Classification

Primary 60J10 Secondary 60G40 05C70 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

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