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Duality Theory for Robust Utility Maximisation

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  • Daniel Bartl
  • Michael Kupper
  • Ariel Neufeld

Abstract

In this paper we present a duality theory for the robust utility maximisation problem in continuous time for utility functions defined on the positive real axis. Our results are inspired by -- and can be seen as the robust analogues of -- the seminal work of Kramkov & Schachermayer [18]. Namely, we show that if the set of attainable trading outcomes and the set of pricing measures satisfy a bipolar relation, then the utility maximisation problem is in duality with a conjugate problem. We further discuss the existence of optimal trading strategies. In particular, our general results include the case of logarithmic and power utility, and they apply to drift and volatility uncertainty.

Suggested Citation

  • Daniel Bartl & Michael Kupper & Ariel Neufeld, 2020. "Duality Theory for Robust Utility Maximisation," Papers 2007.08376, arXiv.org, revised Jun 2021.
    • Handle: RePEc:arx:papers:2007.08376
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    References listed on IDEAS

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    1. Anis Matoussi & Dylan Possamaï & Chao Zhou, 2015. "Robust Utility Maximization In Nondominated Models With 2bsde: The Uncertain Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 25(2), pages 258-287, April.
    2. Daniel Bartl & Michael Kupper & Ariel Neufeld, 2020. "Pathwise superhedging on prediction sets," Finance and Stochastics, Springer, vol. 24(1), pages 215-248, January.
    3. Ariel Neufeld & Marcel Nutz, 2012. "Superreplication under Volatility Uncertainty for Measurable Claims," Papers 1208.6486, arXiv.org, revised Apr 2013.
    4. Marcel Nutz, 2014. "Superreplication under model uncertainty in discrete time," Finance and Stochastics, Springer, vol. 18(4), pages 791-803, October.
    5. Ariel Neufeld & Mario Sikic, 2016. "Robust Utility Maximization in Discrete-Time Markets with Friction," Papers 1610.09230, arXiv.org, revised May 2018.
    6. Amine Ismail & Huyên Pham, 2019. "Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 174-207, January.
    7. Daniel Bartl & Michael Kupper & David J. Promel & Ludovic Tangpi, 2017. "Duality for pathwise superhedging in continuous time," Papers 1705.02933, arXiv.org, revised Apr 2019.
    8. Ariel Neufeld & Mario Šikić, 2019. "Nonconcave robust optimization with discrete strategies under Knightian uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(2), pages 229-253, October.
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    10. Lin, Qian & Sun, Xianming & Zhou, Chao, 2020. "Horizon-unbiased investment with ambiguity," Journal of Economic Dynamics and Control, Elsevier, vol. 114(C).
    11. Daniel Bartl, 2016. "Exponential utility maximization under model uncertainty for unbounded endowments," Papers 1610.00999, arXiv.org, revised Feb 2019.
    12. Lin, Qian & Riedel, Frank, 2014. "Optimal consumption and portfolio choice with ambiguity," Center for Mathematical Economics Working Papers 497, Center for Mathematical Economics, Bielefeld University.
    13. Marcel Nutz & H. Mete Soner, 2010. "Superhedging and Dynamic Risk Measures under Volatility Uncertainty," Papers 1011.2958, arXiv.org, revised Jun 2012.
    14. Marcel Nutz, 2014. "Robust Superhedging with Jumps and Diffusion," Papers 1407.1674, arXiv.org, revised Jul 2015.
    15. Nutz, Marcel, 2015. "Robust superhedging with jumps and diffusion," Stochastic Processes and their Applications, Elsevier, vol. 125(12), pages 4543-4555.
    16. Huyen Pham & Xiaoli Wei & Chao Zhou, 2018. "Portfolio diversification and model uncertainty: a robust dynamic mean-variance approach," Papers 1809.01464, arXiv.org, revised Dec 2021.
    17. Ariel Neufeld & Mario Sikic, 2017. "Nonconcave Robust Optimization with Discrete Strategies under Knightian Uncertainty," Papers 1711.03875, arXiv.org, revised Apr 2019.
    18. Marcel Nutz, 2013. "Superreplication under Model Uncertainty in Discrete Time," Papers 1301.3227, arXiv.org, revised Feb 2014.
    19. Daniel Bartl & Michael Kupper & David J. Prömel & Ludovic Tangpi, 2019. "Duality for pathwise superhedging in continuous time," Finance and Stochastics, Springer, vol. 23(3), pages 697-728, July.
    20. Zhou Yang & Gechun Liang & Chao Zhou, 2017. "Constrained portfolio-consumption strategies with uncertain parameters and borrowing costs," Papers 1711.02939, arXiv.org, revised Dec 2018.
    21. Ariel Neufeld & Marcel Nutz, 2018. "Robust Utility Maximization With Lã‰Vy Processes," Mathematical Finance, Wiley Blackwell, vol. 28(1), pages 82-105, January.
    22. Neufeld, Ariel & Nutz, Marcel, 2014. "Measurability of semimartingale characteristics with respect to the probability law," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3819-3845.
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    Cited by:

    1. Felix-Benedikt Liebrich & Marco Maggis & Gregor Svindland, 2020. "Model Uncertainty: A Reverse Approach," Papers 2004.06636, arXiv.org, revised Mar 2022.
    2. Christoph Czichowsky & Raphael Huwyler, 2022. "Robust utility maximisation under proportional transaction costs for c\`adl\`ag price processes," Papers 2211.00532, arXiv.org, revised May 2023.
    3. Daniel Bartl & Ariel Neufeld & Kyunghyun Park, 2023. "Sensitivity of robust optimization problems under drift and volatility uncertainty," Papers 2311.11248, arXiv.org.
    4. Ariel Neufeld & Julian Sester & Mario Šikić, 2023. "Markov decision processes under model uncertainty," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 618-665, July.
    5. David Criens & Lars Niemann, 2022. "Robust utility maximization with nonlinear continuous semimartingales," Papers 2206.14015, arXiv.org, revised Aug 2023.
    6. Guohui Guan & Zongxia Liang & Yilun Song, 2022. "The continuous-time pre-commitment KMM problem in incomplete markets," Papers 2210.13833, arXiv.org, revised Feb 2023.
    7. Park, Kyunghyun & Wong, Hoi Ying & Yan, Tingjin, 2023. "Robust retirement and life insurance with inflation risk and model ambiguity," Insurance: Mathematics and Economics, Elsevier, vol. 110(C), pages 1-30.
    8. David Criens & Lars Niemann, 2023. "Robust utility maximization with nonlinear continuous semimartingales," Mathematics and Financial Economics, Springer, volume 17, number 5, June.

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