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On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets

Author

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  • Dmitry Kramkov
  • Mihai S^{{i}}rbu

Abstract

We study the two-times differentiability of the value functions of the primal and dual optimization problems that appear in the setting of expected utility maximization in incomplete markets. We also study the differentiability of the solutions to these problems with respect to their initial values. We show that the key conditions for the results to hold true are that the relative risk aversion coefficient of the utility function is uniformly bounded away from zero and infinity, and that the prices of traded securities are sigma-bounded under the num\'{e}raire given by the optimal wealth process.

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  • Dmitry Kramkov & Mihai S^{{i}}rbu, 2006. "On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets," Papers math/0610224, arXiv.org.
    • Handle: RePEc:arx:papers:math/0610224
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    Cited by:

    1. Dmitry Kramkov & Sergio Pulido, 2019. "Density of the set of probability measures with the martingale representation property," Post-Print hal-01598651, HAL.
    2. Dmitry Kramkov & Kim Weston, 2015. "Muckenhoupt's $(A_p)$ condition and the existence of the optimal martingale measure," Papers 1507.05865, arXiv.org.
    3. Mostovyi, Oleksii, 2020. "Asymptotic analysis of the expected utility maximization problem with respect to perturbations of the numéraire," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4444-4469.
    4. Constantinos Kardaras, 2011. "On the closure in the Emery topology of semimartingale wealth-process sets," Papers 1108.0945, arXiv.org, revised Jul 2013.
    5. Sarah Boese & Tracy Cui & Samuel Johnston & Gianmarco Molino & Oleksii Mostovyi, 2020. "Stability and asymptotic analysis of the F\"ollmer-Schweizer decomposition on a finite probability space," Papers 2002.03286, arXiv.org, revised Jun 2020.
    6. Kasper Larsen & Oleksii Mostovyi & Gordan Žitković, 2018. "An expansion in the model space in the context of utility maximization," Finance and Stochastics, Springer, vol. 22(2), pages 297-326, April.
    7. Julio Backhoff Veraguas & Francisco Silva, 2015. "Sensitivity analysis for expected utility maximization in incomplete Brownian market models," Papers 1504.02734, arXiv.org, revised Feb 2017.
    8. M. Mania & R. Tevzadze, 2008. "Backward Stochastic PDEs related to the utility maximization problem," Papers 0806.0240, arXiv.org.
    9. Kramkov, Dmitry & Weston, Kim, 2016. "Muckenhoupt’s (Ap) condition and the existence of the optimal martingale measure," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2615-2633.
    10. Paolo Guasoni & Constantinos Kardaras & Scott Robertson & Hao Xing, 2014. "Abstract, classic, and explicit turnpikes," Finance and Stochastics, Springer, vol. 18(1), pages 75-114, January.
    11. Kasper Larsen & Oleksii Mostovyi & Gordan v{Z}itkovi'c, 2014. "An expansion in the model space in the context of utility maximization," Papers 1410.0946, arXiv.org, revised Aug 2016.
    12. Sergey Nadtochiy & Michael Tehranchi, 2013. "Optimal investment for all time horizons and Martin boundary of space-time diffusions," Papers 1308.2254, arXiv.org, revised Jan 2014.
    13. Anastasiya Tanana, 2023. "Relative performance criteria of multiplicative form in complete markets," Papers 2303.07941, arXiv.org.
    14. Dmitry Kramkov & Mihai S^{{i}}rbu, 2007. "Sensitivity analysis of utility-based prices and risk-tolerance wealth processes," Papers math/0702413, arXiv.org.
    15. Jan Kallsen & Johannes Muhle-Karbe, 2013. "The General Structure of Optimal Investment and Consumption with Small Transaction Costs," Papers 1303.3148, arXiv.org, revised May 2015.
    16. Dmitry Kramkov & Sergio Pulido, 2017. "Density of the set of probability measures with the martingale representation property," Working Papers hal-01598651, HAL.
    17. Kardaras, Constantinos, 2013. "On the closure in the Emery topology of semimartingale wealth-process sets," LSE Research Online Documents on Economics 44996, London School of Economics and Political Science, LSE Library.
    18. Oleksii Mostovyi & Mihai Sîrbu, 2019. "Sensitivity analysis of the utility maximisation problem with respect to model perturbations," Finance and Stochastics, Springer, vol. 23(3), pages 595-640, July.
    19. Dmitry Kramkov & Sergio Pulido, 2017. "Density of the set of probability measures with the martingale representation property," Papers 1709.07329, arXiv.org, revised Jul 2019.
    20. Oleksii Mostovyi & Mihai S^irbu, 2017. "Sensitivity analysis of the utility maximization problem with respect to model perturbations," Papers 1705.08291, arXiv.org.
    21. Michael Mania & Revaz Tevzadze, 2016. "On regularity of primal and dual dynamic value functions related to investment problem," Papers 1604.00525, arXiv.org.
    22. David German, 2010. "Overview of utility-based valuation," Papers 1003.5712, arXiv.org.
    23. Kramkov, D. & Sîrbu, M., 2007. "Asymptotic analysis of utility-based hedging strategies for small number of contingent claims," Stochastic Processes and their Applications, Elsevier, vol. 117(11), pages 1606-1620, November.

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