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Distribution‐constrained optimal stopping

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  • Erhan Bayraktar
  • Christopher W. Miller

Abstract

We solve the problem of optimal stopping of a Brownian motion subject to the constraint that the stopping time's distribution is a given measure consisting of finitely many atoms. In particular, we show that this problem can be converted to a finite sequence of state‐constrained optimal control problems with additional states corresponding to the conditional probability of stopping at each possible terminal time. The proof of this correspondence relies on a new variation of the dynamic programming principle for state‐constrained problems, which avoids measurable selections. We emphasize that distribution constraints lead to novel and interesting mathematical problems on their own, but also demonstrate an application in mathematical finance to model‐free superhedging with an outlook on volatility.

Suggested Citation

  • Erhan Bayraktar & Christopher W. Miller, 2019. "Distribution‐constrained optimal stopping," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 368-406, January.
    • Handle: RePEc:bla:mathfi:v:29:y:2019:i:1:p:368-406
    DOI: 10.1111/mafi.12171
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    References listed on IDEAS

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    1. J. Frederic Bonnans & Xiaolu Tan, 2013. "A model-free no-arbitrage price bound for variance options," Post-Print inria-00634387, HAL.
    2. Bruno Bouchard & Marcel Nutz, 2011. "Weak Dynamic Programming for Generalized State Constraints," Papers 1105.0745, arXiv.org, revised Oct 2012.
    3. A. Galichon & P. Henry-Labord`ere & N. Touzi, 2014. "A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options," Papers 1401.3921, arXiv.org.
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    Cited by:

    1. Sigrid Kallblad, 2017. "A Dynamic Programming Principle for Distribution-Constrained Optimal Stopping," Papers 1703.08534, arXiv.org.
    2. Shantanu Awasthi & Indranil SenGupta, 2020. "First exit-time analysis for an approximate Barndorff-Nielsen and Shephard model with stationary self-decomposable variance process," Papers 2006.07167, arXiv.org, revised Jan 2021.
    3. Christopher W. Miller, 2016. "A Duality Result for Robust Optimization with Expectation Constraints," Papers 1610.01227, arXiv.org.
    4. Soren Christensen & Kristoffer Lindensjo, 2019. "Moment constrained optimal dividends: precommitment \& consistent planning," Papers 1909.10749, arXiv.org.

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