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Robust utility maximization in a stochastic factor model

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  • Hernández-Hernández, Daniel
  • Schied, Alexander

Abstract

We give an explicit PDE characterization for the solution of a robust utility maximization problem in an incomplete market model, whose volatility, interest rate process, 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 negative risk aversion 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. Our method combines two recent advances in the theory of optimal investments: the general duality theory for robust utility maximization and the stochastic control approach to the dual problem of determining optimal martingale measures.

Suggested Citation

  • Hernández-Hernández, Daniel & Schied, Alexander, 2005. "Robust utility maximization in a stochastic factor model," SFB 649 Discussion Papers 2006-007, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    • Handle: RePEc:zbw:sfb649:sfb649dp2006-007
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    References listed on IDEAS

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    1. Lars Peter Hansen & Thomas J Sargent, 2014. "Robust Control and Model Uncertainty," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 5, pages 145-154, World Scientific Publishing Co. Pte. Ltd..
    2. Schied, Alexander, 2005. "Optimal investments for risk- and ambiguity-averse preferences: A duality approach," SFB 649 Discussion Papers 2005-051, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    3. Epstein, Larry G. & Schneider, Martin, 2003. "Recursive multiple-priors," Journal of Economic Theory, Elsevier, vol. 113(1), pages 1-31, November.
    4. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    5. Schied, Alexander & Wu, Ching-Tang, 2005. "Duality theory for optimal investments under model uncertainty," SFB 649 Discussion Papers 2005-025, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    6. Burgert Christian & Rüschendorf Ludger, 2005. "Optimal consumption strategies under model uncertainty," Statistics & Risk Modeling, De Gruyter, vol. 23(1), pages 1-14, January.
    7. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    8. 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.
    9. Schied Alexander & Wu Ching-Tang, 2005. "Duality theory for optimal investments under model uncertainty," Statistics & Risk Modeling, De Gruyter, vol. 23(3), pages 199-217, March.
    10. Burgert Christian & Rüschendorf Ludger, 2005. "Optimal consumption strategies under model uncertainty," Statistics & Risk Modeling, De Gruyter, vol. 23(1/2005), pages 1-14, January.
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    Citations

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    Cited by:

    1. Belkacem Berdjane & Sergei Pergamenshchikov, 2012. "Sequential $\delta$-optimal consumption and investment for stochastic volatility markets with unknown parameters," Papers 1210.5111, arXiv.org, revised May 2015.
    2. Kerem Ugurlu, 2019. "Robust Utility Maximization with Drift and Volatility Uncertainty," Papers 1909.05335, arXiv.org.
    3. repec:hum:wpaper:sfb649dp2007-026 is not listed on IDEAS
    4. Schied, Alexander, 2007. "Robust optimal control for a consumption-investment problem," SFB 649 Discussion Papers 2007-026, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. Kerem Ugurlu, 2018. "Portfolio Optimization with Nondominated Priors and Unbounded Parameters," Papers 1807.05773, arXiv.org.
    6. Wittmüß, Wiebke, 2006. "Robust optimization of consumption with random endowment," SFB 649 Discussion Papers 2006-063, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    7. 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.
    8. Thomas Knispel, 2012. "Asymptotics of robust utility maximization," Papers 1203.1191, arXiv.org.
    9. repec:hum:wpaper:sfb649dp2006-063 is not listed on IDEAS
    10. Hernández-Hernández, Daniel & Schied, Alexander, 2007. "Robust maximization of consumption with logarithmic utility," SFB 649 Discussion Papers 2007-030, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    11. Owari, Keita & 尾張, 圭太, 2008. "Robust Exponential Hedging and Indifference Valuation," Discussion Papers 2008-09, Graduate School of Economics, Hitotsubashi University.
    12. Berdjane Belkacem & Serguei Pergamenchtchikov, 2011. "Optimal consumption and investment for markets with random coefficients," Papers 1102.1186, arXiv.org, revised Dec 2011.
    13. Sara Biagini & Mustafa Pinar, 2015. "The Robust Merton Problem of an Ambiguity Averse Investor," Papers 1502.02847, arXiv.org.
    14. Keita Owari, 2009. "Robust Exponential Hedging in a Brownian Setting," Global COE Hi-Stat Discussion Paper Series gd09-082, Institute of Economic Research, Hitotsubashi University.
    15. Hernández-Hernández, Daniel & Schied, Alexander, 2006. "A control approach to robust utility maximization with logarithmic utility and time-consistent penalties," SFB 649 Discussion Papers 2006-061, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    16. repec:hum:wpaper:sfb649dp2006-061 is not listed on IDEAS
    17. Sigrid Kallblad, 2013. "Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals," Papers 1311.7419, arXiv.org.
    18. Sigrid Kallblad & Jan Obloj & Thaleia Zariphopoulou, 2013. "Time--consistent investment under model uncertainty: the robust forward criteria," Papers 1311.3529, arXiv.org, revised Nov 2014.
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