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Embedding of Walsh Brownian motion

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  • Bayraktar, Erhan
  • Zhang, Xin

Abstract

Let (Z,κ) be a Walsh Brownian motion with spinning measure κ. Suppose μ is a probability measure on Rn. We first provide a necessary and sufficient condition for μ to be a stopping distribution of (Z,κ). Then if the stopped process is required to be uniformly integrable, we show that such a stopping time exists if and only if μ is balanced. Next, under the assumption of being balanced, we identify the minimal stopping times with those τ such that the stopped process Zτ is uniformly integrable. Finally, we generalize Vallois’ embedding, and prove that it minimizes the expectation E[Ψ(LτZ)] among all the admissible solutions τ, where Ψ is a strictly convex function and (LtZ)t≥0 is the local time of the Walsh Brownian motion at the origin.

Suggested Citation

  • Bayraktar, Erhan & Zhang, Xin, 2021. "Embedding of Walsh Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 1-28.
    • Handle: RePEc:eee:spapps:v:134:y:2021:i:c:p:1-28
    DOI: 10.1016/j.spa.2020.10.010
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    References listed on IDEAS

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    1. Vallois, P., 1992. "Quelques inégalités avec le temps local en zero du mouvement Brownien," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 117-155, May.
    2. Karatzas, Ioannis & Yan, Minghan, 2019. "Semimartingales on rays, Walsh diffusions, and related problems of control and stopping," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1921-1963.
    3. Mathias Beiglbock & Marcel Nutz & Florian Stebegg, 2019. "Fine Properties of the Optimal Skorokhod Embedding Problem," Papers 1903.03887, arXiv.org, revised Apr 2020.
    4. A. M. G. Cox & David Hobson & Jan Ob{l}'oj, 2007. "Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping," Papers math/0702173, arXiv.org, revised Nov 2008.
    5. Julien Claisse & Gaoyue Guo & Pierre Henry-Labordère, 2018. "Some Results on Skorokhod Embedding and Robust Hedging with Local Time," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 569-597, November.
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