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Two explicit Skorokhod embeddings for simple symmetric random walk

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  • He, Xue Dong
  • Hu, Sang
  • Obłój, Jan
  • Zhou, Xun Yu

Abstract

Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centred distribution μ on integers into a simple symmetric random walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azéma–Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of μ.

Suggested Citation

  • He, Xue Dong & Hu, Sang & Obłój, Jan & Zhou, Xun Yu, 2019. "Two explicit Skorokhod embeddings for simple symmetric random walk," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3431-3445.
  • Handle: RePEc:eee:spapps:v:129:y:2019:i:9:p:3431-3445
    DOI: 10.1016/j.spa.2018.09.013
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    References listed on IDEAS

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    3. Laurent Carraro & Nicole El Karoui & Jan Ob{l}'oj, 2009. "On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation," Papers 0902.1328, arXiv.org, revised Sep 2012.
    4. Nicholas Barberis, 2012. "A Model of Casino Gambling," Management Science, INFORMS, vol. 58(1), pages 35-51, January.
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