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Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation

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  • Salvatore Federico
  • Paul Gassiat
  • Fausto Gozzi

Abstract

We consider a utility maximization problem for an investment-consumption portfolio when the current utility depends also on the wealth process. Such kind of problems arise, e.g., in portfolio optimization with random horizon or with random trading times. To overcome the difficulties of the problem we use the dual approach. We define a dual problem and treat it by means of dynamic programming, showing that the viscosity solutions of the associated Hamilton-Jacobi-Bellman equation belong to a suitable class of smooth functions. This allows to define a smooth solution of the primal Hamilton-Jacobi-Bellman equation, proving that this solution is indeed unique in a suitable class and coincides with the value function of the primal problem. Some financial applications of the results are provided.

Suggested Citation

  • Salvatore Federico & Paul Gassiat & Fausto Gozzi, 2013. "Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation," Papers 1301.0280, arXiv.org, revised Feb 2015.
    • Handle: RePEc:arx:papers:1301.0280
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    Cited by:

    1. Kexin Chen & Hoi Ying Wong, 2024. "Duality in optimal consumption–investment problems with alternative data," Finance and Stochastics, Springer, vol. 28(3), pages 709-758, July.
    2. Alain Bensoussan & Ka Chun Cheung & Yiqun Li & Sheung Chi Phillip Yam, 2022. "Inter‐temporal mutual‐fund management," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 825-877, July.
    3. Salvatore Federico & Paul Gassiat, 2014. "Viscosity Characterization of the Value Function of an Investment-Consumption Problem in Presence of an Illiquid Asset," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 966-991, March.
    4. Kexin Chen & Hoi Ying Wong, 2022. "Duality in optimal consumption--investment problems with alternative data," Papers 2210.08422, arXiv.org, revised Jul 2023.
    5. Daniel Sevcovic & Cyril Izuchukwu Udeani, 2021. "Application of maximal monotone operator method for solving Hamilton-Jacobi-Bellman equation arising from optimal portfolio selection problem," Papers 2104.06115, arXiv.org.
    6. Ashley Davey & Michael Monoyios & Harry Zheng, 2021. "Duality for optimal consumption with randomly terminating income," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1275-1314, October.
    7. Michael Monoyios, 2020. "Infinite horizon utility maximisation from inter-temporal wealth," Papers 2009.00972, arXiv.org, revised Oct 2020.
    8. Agostino Capponi & Lijun Bo, 2016. "Robust Optimization of Credit Portfolios," Papers 1603.08169, arXiv.org.
    9. Filippo de Feo & Salvatore Federico & Andrzej 'Swik{e}ch, 2023. "Optimal control of stochastic delay differential equations and applications to path-dependent financial and economic models," Papers 2302.08809, arXiv.org.
    10. Dadashi, Hassan, 2020. "Optimal investment–consumption problem: Post-retirement with minimum guarantee," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 160-181.
    11. Lijun Bo & Agostino Capponi, 2017. "Robust Optimization of Credit Portfolios," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 30-56, January.
    12. Jose Cruz & Maria Grossinho & Daniel Sevcovic & Cyril Izuchukwu Udeani, 2022. "Linear and Nonlinear Partial Integro-Differential Equations arising from Finance," Papers 2207.11568, arXiv.org.

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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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