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Duality theory for optimal investments under model uncertainty

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  • Alexander Schied
  • Ching-Tang Wu

Abstract

Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this paper, we study the duality theory for the problem of maximizing the robust utility of the terminal wealth in a general incomplete market model. We also allow for very general sets of prior models. In particular, we do not assume that all prior models are equivalent to each other, which allows us to handle many economically meaningful robust utility functionals such as those defined by AVaR(lambda), concave distortions, or convex capacities. We also show that dropping the equivalence of prior models may lead to new effects such as the existence of arbitrage strategies under the least favorable model.

Suggested Citation

  • 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.
  • Handle: RePEc:hum:wpaper:sfb649dp2005-025
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    References listed on IDEAS

    as
    1. Alexander Schied, 2004. "On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals," Papers math/0407127, arXiv.org.
    2. 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.
    3. Thomas Goll & Ludger Rüschendorf, 2001. "Minimax and minimal distance martingale measures and their relationship to portfolio optimization," Finance and Stochastics, Springer, vol. 5(4), pages 557-581.
    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. L. Randall Wray & Stephanie Bell, 2004. "Introduction," Chapters,in: Credit and State Theories of Money, chapter 1 Edward Elgar Publishing.
    6. Gilboa, Itzhak, 1987. "Expected utility with purely subjective non-additive probabilities," Journal of Mathematical Economics, Elsevier, vol. 16(1), pages 65-88, February.
    7. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    8. Philippe Robert-Demontrond & R. Ringoot, 2004. "Introduction," Post-Print halshs-00081823, HAL.
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    Citations

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    Cited by:

    1. 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.
    2. Alexander Schied, 2007. "Robust Optimal Control for a Consumption-investment Problem," SFB 649 Discussion Papers SFB649DP2007-026, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    3. Keita Owari, 2011. "On Admissible Strategies in Robust Utility Maximization," Papers 1109.5512, arXiv.org, revised Mar 2012.
    4. 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.
    5. Trevino Aguilar Erick, 2009. "Robust efficient hedging for American options: The existence of worst case probability measures," Statistics & Risk Modeling, De Gruyter, vol. 27(1), pages 1-23, November.
    6. repec:wsi:ijtafx:v:20:y:2017:i:03:n:s0219024917500157 is not listed on IDEAS
    7. Keita Owari, 2013. "A Robust Version of Convex Integral Functionals," Papers 1305.6023, arXiv.org, revised May 2015.
    8. repec:spr:compst:v:67:y:2008:i:1:p:1-20 is not listed on IDEAS
    9. Gundel, Anne & Weber, Stefan, 2008. "Utility maximization under a shortfall risk constraint," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1126-1151, December.
    10. 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.
    11. Thomas Knispel, 2012. "Asymptotics of robust utility maximization," Papers 1203.1191, arXiv.org.
    12. Owari, Keita, 2008. "Robust Exponential Hedging and Indifference Valuation," Discussion Papers 2008-09, Graduate School of Economics, Hitotsubashi University.
    13. Anis Matoussi & Hanen Mezghani & Mohamed Mnif, 2013. "Maximization of recursive utilities under convex portfolio constraints," Papers 1307.0872, arXiv.org, revised Sep 2014.
    14. Keita Owari, 2011. "ON ADMISSIBLE STRATEGIES IN ROBUST UTILITY MAXIMIZATION(Revised in March 2012, Forthcoming in "Mathematics and Financial Economics")," CARF F-Series CARF-F-257, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    15. Constantinos Kardaras & Scott Robertson, 2010. "Robust maximization of asymptotic growth," Papers 1005.3454, arXiv.org, revised Aug 2012.
    16. 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.
    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.
    19. Hernández-Hernández, Daniel & Schied, Alexander, 2007. "A control approach to robust utility maximization with logarithmic utility and time-consistent penalties," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 980-1000, August.
    20. 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.
    21. Kardaras, Constantinos & Robertson, Scott, 2012. "Robust maximization of asymptotic growth," LSE Research Online Documents on Economics 44994, London School of Economics and Political Science, LSE Library.

    More about this item

    Keywords

    model uncertainty; duality theory; investment; uncertainty; utility; arbitrage;

    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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