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Robust optimal control for a consumption-investment problem

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  • Alexander Schied

Abstract

We give an explicit PDE characterization for the solution of the problem of maximizing the utility of both terminal wealth and intertemporal consumption under model uncertainty. The underlying market model consists of a risky asset, whose volatility and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with risk aversion parameter 0 > α > 1 and a dynamically consistent coherent risk measure, which allows for model uncertainty in the distributions of both the asset price dynamics and the factor process. Ourmethod combines recent results by Wittmüß (Robust optimization of consumption with random endowment, 2006) on the duality theory of robust optimization of consumption with a stochastic control approach to the dual problem of determining a ‘worst-case martingale measure’. Copyright Springer-Verlag 2008

Suggested Citation

  • Alexander Schied, 2008. "Robust optimal control for a consumption-investment problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(1), pages 1-20, February.
    • Handle: RePEc:spr:mathme:v:67:y:2008:i:1:p:1-20
    DOI: 10.1007/s00186-007-0172-y
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    References listed on IDEAS

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    1. Alexander Schied & Ching-Tang Wu, 2005. "Duality theory for optimal investments under model uncertainty," SFB 649 Discussion Papers SFB649DP2005-025, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany, revised Sep 2005.
    2. Alexander Schied, 2005. "Optimal Investments for Robust Utility Functionals in Complete Market Models," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 750-764, August.
    3. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    4. Ralf Korn & Olaf Menkens, 2005. "Worst-Case Scenario Portfolio Optimization: a New Stochastic Control Approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 62(1), pages 123-140, September.
    5. Anne Gundel, 2005. "Robust utility maximization for complete and incomplete market models," Finance and Stochastics, Springer, vol. 9(2), pages 151-176, April.
    6. Alexander Schied, 2007. "Optimal investments for risk- and ambiguity-averse preferences: a duality approach," Finance and Stochastics, Springer, vol. 11(1), pages 107-129, January.
    7. Burgert Christian & Rüschendorf Ludger, 2005. "Optimal consumption strategies under model uncertainty," Statistics & Risk Modeling, De Gruyter, vol. 23(1), pages 1-14, January.
    8. Denis Talay & Ziyu Zheng, 2002. "Worst case model risk management," Finance and Stochastics, Springer, vol. 6(4), pages 517-537.
    9. Ralf Korn & Paul Wilmott, 2002. "Optimal Portfolios Under The Threat Of A Crash," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 171-187.
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    Cited by:

    1. Qian Lin & Frank Riedel, 2021. "Optimal consumption and portfolio choice with ambiguous interest rates and volatility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 1189-1202, April.
    2. Dariusz Zawisza, 2016. "Smooth solutions to discounted reward control problems with unbounded discount rate and financial applications," Papers 1602.00899, arXiv.org, revised Feb 2016.
    3. José López-Barrientos & Héctor Jasso-Fuentes & Beatris Escobedo-Trujillo, 2015. "Discounted robust control for Markov diffusion processes," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(1), pages 53-76, April.
    4. Dirk Becherer & Klebert Kentia, 2017. "Hedging under generalized good-deal bounds and model uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 171-214, August.
    5. Wing Fung Chong & Gechun Liang, 2018. "Optimal investment and consumption with forward preferences and uncertain parameters," Papers 1807.01186, arXiv.org, revised Nov 2023.
    6. Dariusz Zawisza, 2020. "A note on the worst case approach for a market with a stochastic interest rate," Papers 2001.01998, arXiv.org.
    7. Revaz Tevzadze & Teimuraz Toronjadze & Tamaz Uzunashvili, 2013. "Robust utility maximization for a diffusion market model with misspecified coefficients," Finance and Stochastics, Springer, vol. 17(3), pages 535-563, July.
    8. Frank Thomas Seifried, 2010. "Optimal Investment for Worst-Case Crash Scenarios: A Martingale Approach," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 559-579, August.
    9. Wahid Faidi & Anis Matoussi & Mohamed Mnif, 2017. "Optimal Stochastic Control Problem Under Model Uncertainty With Nonentropy Penalty," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-41, May.
    10. Zongxia Liang & Ming Ma, 2020. "Robust consumption‐investment problem under CRRA and CARA utilities with time‐varying confidence sets," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 1035-1072, July.
    11. Thomas Knispel, 2012. "Asymptotics of robust utility maximization," Papers 1203.1191, arXiv.org.
    12. Bogdan Iftimie, 2023. "A robust investment-consumption optimization problem in a switching regime interest rate setting," Journal of Global Optimization, Springer, vol. 86(3), pages 713-739, July.
    13. 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.
    14. Dirk Becherer & Klebert Kentia, 2016. "Hedging under generalized good-deal bounds and model uncertainty," Papers 1607.04488, arXiv.org, revised Apr 2017.
    15. 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.

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