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From optimal martingales to randomized dual optimal stopping

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  • Denis Belomestny
  • John Schoenmakers

Abstract

In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the surely optimal Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense robust under a particular random perturbation. This new insight leads to a novel randomized dual martingale minimization algorithm that doesn't require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm may efficiently select a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance.

Suggested Citation

  • Denis Belomestny & John Schoenmakers, 2023. "From optimal martingales to randomized dual optimal stopping," Quantitative Finance, Taylor & Francis Journals, vol. 23(7-8), pages 1099-1113, August.
    • Handle: RePEc:taf:quantf:v:23:y:2023:i:7-8:p:1099-1113
    DOI: 10.1080/14697688.2023.2223242
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    Cited by:

    1. Christian Bayer & Luca Pelizzari & John Schoenmakers, 2023. "Primal and dual optimal stopping with signatures," Papers 2312.03444, arXiv.org.

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