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An analytical solution for the HJB equation arising from the Merton problem

Author

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  • Song-Ping Zhu

    (School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia)

  • Guiyuan Ma

    (School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia)

Abstract

In this paper, an analytical solution for the well-known Hamilton–Jacobi–Bellman (HJB) equation that arises from the Merton problem subject to general utility functions is presented for the first time. The solution presented in this paper is written in the form of a Taylor’s series expansion and constructed through the homotopy analysis method (HAM). The fully nonlinear HJB equation is decomposed into an infinite series of linear PDEs which can be solved analytically. Four examples are presented with the first two cases showing the accuracy of our solution approach; while the last two demonstrating its versatility.

Suggested Citation

  • Song-Ping Zhu & Guiyuan Ma, 2018. "An analytical solution for the HJB equation arising from the Merton problem," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(01), pages 1-26, March.
    • Handle: RePEc:wsi:ijfexx:v:05:y:2018:i:01:n:s2424786318500081
    DOI: 10.1142/S2424786318500081
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    References listed on IDEAS

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    Cited by:

    1. Guiyuan Ma & Song-Ping Zhu & Boda Kang, 2020. "A Numerical Solution of Optimal Portfolio Selection Problem with General Utility Functions," Computational Economics, Springer;Society for Computational Economics, vol. 55(3), pages 957-981, March.
    2. Guiyuan Ma & Song-Ping Zhu, 2022. "Revisiting the Merton Problem: from HARA to CARA Utility," Computational Economics, Springer;Society for Computational Economics, vol. 59(2), pages 651-686, February.
    3. Hidekazu Yoshioka & Yuta Yaegashi, 2020. "A growth rate control problem of harmful species population and its application to algae bloom," Environment Systems and Decisions, Springer, vol. 40(1), pages 107-124, March.
    4. Christelle Dleuna Nyoumbi & Antoine Tambue, 2023. "A Novel High Dimensional Fitted Scheme for Stochastic Optimal Control Problems," Computational Economics, Springer;Society for Computational Economics, vol. 61(1), pages 1-34, January.

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