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Dual formulation of the utility maximization problem: the case of nonsmooth utility

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  • B. Bouchard
  • N. Touzi
  • A. Zeghal

Abstract

We study the dual formulation of the utility maximization problem in incomplete markets when the utility function is finitely valued on the whole real line. We extend the existing results in this literature in two directions. First, we allow for nonsmooth utility functions, so as to include the shortfall minimization problems in our framework. Second, we allow for the presence of some given liability or a random endowment. In particular, these results provide a dual formulation of the utility indifference valuation rule.

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  • B. Bouchard & N. Touzi & A. Zeghal, 2004. "Dual formulation of the utility maximization problem: the case of nonsmooth utility," Papers math/0405290, arXiv.org.
    • Handle: RePEc:arx:papers:math/0405290
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    References listed on IDEAS

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    1. Yuri M. Kabanov & Christophe Stricker, 2002. "On the optimal portfolio for the exponential utility maximization: remarks to the six‐author paper," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 125-134, April.
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    Cited by:

    1. Westray, Nicholas & Zheng, Harry, 2009. "Constrained nonsmooth utility maximization without quadratic inf convolution," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1561-1579, May.
    2. Carole Bernard & Stephan Sturm, 2022. "Cost-efficiency in Incomplete Markets," Papers 2206.12511, arXiv.org, revised Jul 2024.
    3. Guasoni, Paolo & Muhle-Karbe, Johannes & Xing, Hao, 2017. "Robust portfolios and weak incentives in long-run investments," LSE Research Online Documents on Economics 60577, London School of Economics and Political Science, LSE Library.
    4. Paolo Guasoni & Johannes Muhle-Karbe & Hao Xing, 2013. "Robust Portfolios and Weak Incentives in Long-Run Investments," Papers 1306.2751, arXiv.org, revised Aug 2014.
    5. Miklos Rasonyi & Lukasz Stettner, 2005. "On utility maximization in discrete-time financial market models," Papers math/0505243, arXiv.org.
    6. Kasper Larsen & Gordan v{Z}itkovi'c, 2011. "On utility maximization under convex portfolio constraints," Papers 1102.0346, arXiv.org, revised Feb 2013.
    7. Nuerxiati Abudurexiti & Erhan Bayraktar & Takaki Hayashi & Hasanjan Sayit, 2024. "Two-fund separation under hyperbolically distributed returns and concave utility function," Papers 2410.04459, arXiv.org, revised Jan 2025.
    8. Bruno Bouchard & Huy^en Pham, 2006. "Optimal consumption in discrete-time financial models with industrial investment opportunities and nonlinear returns," Papers math/0602451, arXiv.org.
    9. Würth, Andreas & Schumacher, J.M., 2011. "Risk aversion for nonsmooth utility functions," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 109-128, March.
    10. Marcos Escobar-Anel, 2022. "A dynamic programming approach to path-dependent constrained portfolios," Annals of Operations Research, Springer, vol. 315(1), pages 141-157, August.
    11. Nicholas Westray & Harry Zheng, 2011. "Minimal sufficient conditions for a primal optimizer in nonsmooth utility maximization," Finance and Stochastics, Springer, vol. 15(3), pages 501-512, September.
    12. Maxim Bichuch & Stephan Sturm, 2011. "Portfolio Optimization under Convex Incentive Schemes," Papers 1109.2945, arXiv.org, revised Oct 2013.
    13. Ilhan, Aytaç & Jonsson, Mattias & Sircar, Ronnie, 2009. "Optimal static-dynamic hedges for exotic options under convex risk measures," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3608-3632, October.
    14. Maxim Bichuch & Stephan Sturm, 2014. "Portfolio optimization under convex incentive schemes," Finance and Stochastics, Springer, vol. 18(4), pages 873-915, October.
    15. Marcos Escobar-Anel & Michel Kschonnek & Rudi Zagst, 2022. "Portfolio optimization: not necessarily concave utility and constraints on wealth and allocation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 101-140, February.
    16. Frank Seifried, 2010. "Optimal investment with deferred capital gains taxes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 181-199, February.
    17. Zongxia Liang & Yang Liu & Litian Zhang, 2021. "A Framework of State-dependent Utility Optimization with General Benchmarks," Papers 2101.06675, arXiv.org, revised Dec 2023.
    18. Nicholas Westray & Harry Zheng, 2010. "Constrained NonSmooth Utility Maximization on the Positive Real Line," Papers 1010.4055, arXiv.org.
    19. Miklos Rasonyi, 2017. "On utility maximization without passing by the dual problem," Papers 1702.00982, arXiv.org, revised Mar 2018.

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