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A Generalized Finite Difference Method for Solving Hamilton–Jacobi–Bellman Equations in Optimal Investment

Author

Listed:
  • Jiamian Lin

    (Department of Mathematics, Jinan University, Guangzhou 510632, China)

  • Xi Li

    (Department of Mathematics, Jinan University, Guangzhou 510632, China)

  • SingRu (Celine) Hoe

    (College of Business, Texas A&M University-Commerce, Commerce, TX 75428, USA)

  • Zhongfeng Yan

    (Department of Mathematics, Jinan University, Guangzhou 510632, China)

Abstract

This paper studies the numerical algorithm of stochastic control problems in investment optimization. Investors choose the optimal investment to maximize the expected return under uncertainty. The optimality condition, the Hamilton–Jacobi–Bellman (HJB) equation, satisfied by the value function and obtained by the dynamic programming method, is a partial differential equation coupled with optimization. One of the major computational difficulties is the irregular boundary conditions presented in the HJB equation. In this paper, two mesh-free algorithms are proposed to solve two different cases of HJB equations with regular and irregular boundary conditions. The model of optimal investment under uncertainty developed by Abel is used to study the efficacy of the proposed algorithms. Extensive numerical studies are conducted to test the impact of the key parameters on the numerical efficacy. By comparing the numerical solution with the exact solution, the proposed numerical algorithms are validated.

Suggested Citation

  • Jiamian Lin & Xi Li & SingRu (Celine) Hoe & Zhongfeng Yan, 2023. "A Generalized Finite Difference Method for Solving Hamilton–Jacobi–Bellman Equations in Optimal Investment," Mathematics, MDPI, vol. 11(10), pages 1-20, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2346-:d:1149599
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    References listed on IDEAS

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    2. Wang, J. & Forsyth, P.A., 2010. "Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation," Journal of Economic Dynamics and Control, Elsevier, vol. 34(2), pages 207-230, February.
    3. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    4. Richard Bellman, 1954. "Some Applications of the Theory of Dynamic Programming---A Review," Operations Research, INFORMS, vol. 2(3), pages 275-288, August.
    5. H. Alwardi & S. Wang & L. Jennings & S. Richardson, 2012. "An adaptive least-squares collocation radial basis function method for the HJB equation," Journal of Global Optimization, Springer, vol. 52(2), pages 305-322, February.
    6. Richard Bellman, 1954. "On some applications of the theory of dynamic programming to logistics," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 1(2), pages 141-153, June.
    7. Abel, Andrew B, 1983. "Optimal Investment under Uncertainty," American Economic Review, American Economic Association, vol. 73(1), pages 228-233, March.
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