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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

This paper deals with an investment–consumption portfolio problem when the current utility depends also on the wealth process. Such problems arise e.g. in portfolio optimization with random horizon or random trading times. To overcome the difficulties of the problem, a dual approach is employed: a dual control problem is defined and treated 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 defining a smooth solution of the primal Hamilton–Jacobi–Bellman equation, and proving by verification that such a solution is indeed unique in a suitable class of smooth functions and coincides with the value function of the primal problem. Applications to specific financial problems are given. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Salvatore Federico & Paul Gassiat & Fausto Gozzi, 2015. "Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation," Finance and Stochastics, Springer, vol. 19(2), pages 415-448, April.
    • Handle: RePEc:spr:finsto:v:19:y:2015:i:2:p:415-448
    DOI: 10.1007/s00780-015-0257-z
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    7. 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.
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    11. Bruno Bouchard & Huyên Pham, 2004. "Wealth-path dependent utility maximization in incomplete markets," Finance and Stochastics, Springer, vol. 8(4), pages 579-603, November.
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    Cited by:

    1. 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.
    2. 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.
    3. Agostino Capponi & Lijun Bo, 2016. "Robust Optimization of Credit Portfolios," Papers 1603.08169, arXiv.org.
    4. 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.
    5. Dadashi, Hassan, 2020. "Optimal investment–consumption problem: Post-retirement with minimum guarantee," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 160-181.
    6. 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.
    7. 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.
    8. 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.
    9. Kexin Chen & Hoi Ying Wong, 2022. "Duality in optimal consumption--investment problems with alternative data," Papers 2210.08422, arXiv.org, revised Jul 2023.
    10. Michael Monoyios, 2020. "Infinite horizon utility maximisation from inter-temporal wealth," Papers 2009.00972, arXiv.org, revised Oct 2020.
    11. Lijun Bo & Agostino Capponi, 2017. "Robust Optimization of Credit Portfolios," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 30-56, January.

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    More about this item

    Keywords

    Optimal stochastic control; Investment–consumption problem; Duality; Hamilton–Jacobi–Bellman equation; Regularity of viscosity solutions; 93E20; 49L20; 90C46; 91G80; 35B65; C61; G11;
    All these keywords.

    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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