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A Control Approach to Robust Utility Maximization with Logarithmic Utility and Time-Consistent Penalties

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  • Daniel Hernandez–Hernandez
  • Alexander Schied

Abstract

We propose a stochastic control approach to the dynamic maximization of robust utility functionals that are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. In particular, the market model is incomplete. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions. The fact that we obtain classical solutions rather than viscosity solutions is important for the use of numerical algorithms, whose applicability is demonstrated in examples.

Suggested Citation

  • Daniel Hernandez–Hernandez & Alexander Schied, 2006. "A Control Approach to Robust Utility Maximization with Logarithmic Utility and Time-Consistent Penalties," SFB 649 Discussion Papers SFB649DP2006-061, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    • Handle: RePEc:hum:wpaper:sfb649dp2006-061
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    References listed on IDEAS

    as
    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. Daniel Hernandez–Hernandez & Alexander Schied, 2005. "Robust Utility Maximization in a Stochastic Factor Model," SFB 649 Discussion Papers SFB649DP2006-007, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany, revised Aug 2006.
    3. Hernández-Hernández Daniel & Schied Alexander, 2006. "Robust utility maximization in a stochastic factor model," Statistics & Risk Modeling, De Gruyter, vol. 24(1/2006), pages 1-17, July.
    4. E. Jouini & W. Schachermayer & N. Touzi, 2008. "Optimal Risk Sharing For Law Invariant Monetary Utility Functions," Mathematical Finance, Wiley Blackwell, vol. 18(2), pages 269-292, April.
    5. Hernández-Hernández Daniel & Schied Alexander, 2006. "Robust utility maximization in a stochastic factor model," Statistics & Risk Modeling, De Gruyter, vol. 24(1), pages 109-125, July.
    6. 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.
    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. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    9. 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.
    10. Alexander Schied, 2005. "Optimal Investments for Risk- and Ambiguity-Averse Preferences: A Duality Approach," SFB 649 Discussion Papers SFB649DP2005-051, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany, revised Aug 2006.
    11. Schied Alexander & Wu Ching-Tang, 2005. "Duality theory for optimal investments under model uncertainty," Statistics & Risk Modeling, De Gruyter, vol. 23(3/2005), pages 199-217, March.
    12. 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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    Cited by:

    1. Alexander Schied, 2007. "Robust Optimal Control for a Consumption-investment Problem," SFB 649 Discussion Papers SFB649DP2007-026, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    2. Wiebke Wittmüß, 2006. "Robust Optimization of Consumption with Random Endowment," SFB 649 Discussion Papers SFB649DP2006-063, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Daniel Hernández-Hernández & Alexander Schied, 2007. "Robust Maximization of Consumption with Logarithmic Utility," SFB 649 Discussion Papers SFB649DP2007-030, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.

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    More about this item

    Keywords

    Optimal investment; model uncertainty; incomplete markets; stochastic volatility; coherent risk measure; convex risk measure; optimal control; convex duality;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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