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Weak Dynamic Programming for Generalized State Constraints

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  • Bruno Bouchard
  • Marcel Nutz

Abstract

We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the viscosity sense. We treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints.

Suggested Citation

  • Bruno Bouchard & Marcel Nutz, 2011. "Weak Dynamic Programming for Generalized State Constraints," Papers 1105.0745, arXiv.org, revised Oct 2012.
    • Handle: RePEc:arx:papers:1105.0745
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    Cited by:

    1. Ludovic Moreau & Johannes Muhle-Karbe & H. Mete Soner, 2014. "Trading with Small Price Impact," Papers 1402.5304, arXiv.org, revised Mar 2015.
    2. Marcel Nutz & Johannes Wiesel & Long Zhao, 2022. "Limits of Semistatic Trading Strategies," Papers 2204.12251, arXiv.org.
    3. Romuald Elie & Ludovic Moreau & Dylan Possamai, 2017. "On a class of path-dependent singular stochastic control problems," Papers 1701.08861, arXiv.org, revised Feb 2018.
    4. Christopher W. Miller, 2016. "A Duality Result for Robust Optimization with Expectation Constraints," Papers 1610.01227, arXiv.org.
    5. Sigrid Kallblad, 2017. "A Dynamic Programming Principle for Distribution-Constrained Optimal Stopping," Papers 1703.08534, arXiv.org.
    6. Bruno Bouchard & Ludovic Moreau & Marcel Nutz, 2012. "Stochastic target games with controlled loss," Papers 1206.6325, arXiv.org, revised Apr 2014.
    7. Gordan Zitkovic, 2013. "Dynamic Programming for controlled Markov families: abstractly and over Martingale Measures," Papers 1307.5163, arXiv.org, revised Mar 2014.
    8. Erhan Bayraktar & Christopher W. Miller, 2019. "Distribution‐constrained optimal stopping," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 368-406, January.
    9. Yuk-Loong Chow & Xiang Yu & Chao Zhou, 2020. "On Dynamic Programming Principle for Stochastic Control Under Expectation Constraints," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 803-818, June.
    10. Mourad Lazgham, 2018. "Regularity properties in a state-constrained expected utility maximization problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(2), pages 185-240, October.
    11. Mourad Lazgham, 2015. "Regularity properties in a state-constrained expected utility maximization problem," Papers 1510.03079, arXiv.org.
    12. Bruno Bouchard & Ludovic Moreau & Mete H. Soner, 2013. "Hedging under an expected loss constraint with small transaction costs," Papers 1309.4916, arXiv.org, revised Sep 2014.
    13. Dylan Possamai & Xiaolu Tan & Chao Zhou, 2015. "Stochastic control for a class of nonlinear kernels and applications," Papers 1510.08439, arXiv.org, revised Jul 2017.
    14. Bruno Bouchard & Ludovic Moreau & Mete H. Soner, 2016. "Hedging under an expected loss constraint with small transaction costs," Post-Print hal-00863562, HAL.
    15. Marcel Nutz & Johannes Wiesel & Long Zhao, 2023. "Limits of semistatic trading strategies," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 185-205, January.
    16. Thai Nguyen, 2016. "Optimal investment and consumption with downside risk constraint in jump-diffusion models," Papers 1604.05584, arXiv.org.
    17. Johannes Muhle-Karbe & Max Reppen & H. Mete Soner, 2016. "A Primer on Portfolio Choice with Small Transaction Costs," Papers 1612.01302, arXiv.org, revised May 2017.
    18. Adrien Nguyen Huu & Nadia Oudjane, 2014. "Hedging Expected Losses on Derivatives in Electricity Futures Markets," Papers 1401.8271, arXiv.org.
    19. H. Mete Soner & Mirjana Vukelja, 2016. "Utility maximization in an illiquid market in continuous time," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(2), pages 285-321, October.
    20. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.

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