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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. 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.
    5. 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.
    6. Bruno Bouchard & Ludovic Moreau & Mete H. Soner, 2016. "Hedging under an expected loss constraint with small transaction costs," Post-Print hal-00863562, HAL.
    7. Christopher W. Miller, 2016. "A Duality Result for Robust Optimization with Expectation Constraints," Papers 1610.01227, arXiv.org.
    8. Marcel Nutz & Johannes Wiesel & Long Zhao, 2023. "Limits of semistatic trading strategies," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 185-205, January.
    9. Thai Nguyen, 2016. "Optimal investment and consumption with downside risk constraint in jump-diffusion models," Papers 1604.05584, arXiv.org.
    10. Sigrid Kallblad, 2017. "A Dynamic Programming Principle for Distribution-Constrained Optimal Stopping," Papers 1703.08534, arXiv.org.
    11. Bruno Bouchard & Ludovic Moreau & Marcel Nutz, 2012. "Stochastic target games with controlled loss," Papers 1206.6325, arXiv.org, revised Apr 2014.
    12. Gordan Zitkovic, 2013. "Dynamic Programming for controlled Markov families: abstractly and over Martingale Measures," Papers 1307.5163, arXiv.org, revised Mar 2014.
    13. 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.
    14. Erhan Bayraktar & Christopher W. Miller, 2019. "Distribution‐constrained optimal stopping," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 368-406, January.
    15. Adrien Nguyen Huu & Nadia Oudjane, 2014. "Hedging Expected Losses on Derivatives in Electricity Futures Markets," Papers 1401.8271, arXiv.org.
    16. 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.
    17. 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.
    18. 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.
    19. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    20. Mourad Lazgham, 2015. "Regularity properties in a state-constrained expected utility maximization problem," Papers 1510.03079, arXiv.org.

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