IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i10p2346-d1149599.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/10/2346/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/10/2346/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Waleeda Swaidan & Amran Hussin, 2013. "Feedback Control Method Using Haar Wavelet Operational Matrices for Solving Optimal Control Problems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, August.
    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. 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.
    4. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    5. Richard Bellman, 1954. "Some Applications of the Theory of Dynamic Programming---A Review," Operations Research, INFORMS, vol. 2(3), pages 275-288, August.
    6. Abel, Andrew B, 1983. "Optimal Investment under Uncertainty," American Economic Review, American Economic Association, vol. 73(1), pages 228-233, March.
    7. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Matthias Breuer & David Windisch, 2019. "Investment Dynamics and Earnings‐Return Properties: A Structural Approach," Journal of Accounting Research, Wiley Blackwell, vol. 57(3), pages 639-674, June.
    2. Michael J. Pennock & William B. Rouse & Diane L. Kollar, 2007. "Transforming the Acquisition Enterprise: A Framework for Analysis and a Case Study of Ship Acquisition," Systems Engineering, John Wiley & Sons, vol. 10(2), pages 99-117, June.
    3. Wu, Tong & Lawell, C.Y. Cynthia Lin & Just, David R. & Zhao, Jiancheng & Fei, Zhangjun & Wei, Qiang, 2022. "Optimal Forest Management for Interdependent Products: A Nested Dynamic Bioeconomic Model and Application to Bamboo," 2022 Annual Meeting, July 31-August 2, Anaheim, California 322164, Agricultural and Applied Economics Association.
    4. Xiaoyue Li & John M. Mulvey, 2023. "Optimal Portfolio Execution in a Regime-switching Market with Non-linear Impact Costs: Combining Dynamic Program and Neural Network," Papers 2306.08809, arXiv.org.
    5. Nick Bloom & Stephen Bond & John Van Reenen, 2007. "Uncertainty and Investment Dynamics," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 74(2), pages 391-415.
    6. Mahmoud Mahfouz & Angelos Filos & Cyrine Chtourou & Joshua Lockhart & Samuel Assefa & Manuela Veloso & Danilo Mandic & Tucker Balch, 2019. "On the Importance of Opponent Modeling in Auction Markets," Papers 1911.12816, arXiv.org.
    7. Chen, Cheng & Senga, Tatsuro & Sun, Chang & Zhang, Hongyong, 2023. "Uncertainty, imperfect information, and expectation formation over the firm’s life cycle," Journal of Monetary Economics, Elsevier, vol. 140(C), pages 60-77.
    8. Andrea Bassanini & Stefano Scarpetta, 2003. "The Driving Forces of Economic Growth: Panel Data Evidence for the OECD Countries," OECD Economic Studies, OECD Publishing, vol. 2001(2), pages 9-56.
    9. Mailand, Wilhelm, 1998. "Zum Einfluß von Unsicherheit auf die gesamtwirtschaftliche Investitionstätigkeit," HWWA Discussion Papers 57, Hamburg Institute of International Economics (HWWA).
    10. Kaido Kepp & Kadri Männasoo, 2021. "Investment irreversibility and cyclical adversity: Implications for the financial performance of European manufacturing companies," Managerial and Decision Economics, John Wiley & Sons, Ltd., vol. 42(7), pages 1665-1678, October.
    11. Nicholas Bloom & Max Floetotto & Nir Jaimovich & Itay Saporta†Eksten & Stephen J. Terry, 2018. "Really Uncertain Business Cycles," Econometrica, Econometric Society, vol. 86(3), pages 1031-1065, May.
    12. Goel, Rajeev K. & Mazhar, Ummad & Sayan, Serdar, 2021. "Strategic location of firms: Does it empower bribe givers or bribe takers?," Economic Systems, Elsevier, vol. 45(3).
    13. Klaus Mohn & Petter Osmundsen, 2011. "Asymmetry and uncertainty in capital formation: an application to oil investment," Applied Economics, Taylor & Francis Journals, vol. 43(28), pages 4387-4401.
    14. Joseph P. Byrne & E. Philip Davis, 2005. "Investment and Uncertainty in the G7," Review of World Economics (Weltwirtschaftliches Archiv), Springer;Institut für Weltwirtschaft (Kiel Institute for the World Economy), vol. 141(1), pages 1-32, April.
    15. Panagiotidis, Theodore & Printzis, Panagiotis, 2020. "What is the investment loss due to uncertainty?," Global Finance Journal, Elsevier, vol. 45(C).
    16. Henriques, Irene & Sadorsky, Perry, 2011. "The effect of oil price volatility on strategic investment," Energy Economics, Elsevier, vol. 33(1), pages 79-87, January.
    17. De Sousa, José & Disdier, Anne-Célia & Gaigné, Carl, 2020. "Export decision under risk," European Economic Review, Elsevier, vol. 121(C).
    18. Dawei Chen & Fangxu Mo & Ye Chen & Jun Zhang & Xinyu You, 2022. "Optimization of Ramp Locations along Freeways: A Dynamic Programming Approach," Sustainability, MDPI, vol. 14(15), pages 1-13, August.
    19. Raouf Boucekkine & Bruno de Oliveira Cruz, 2015. "Technological Progress and Investment: A Non-Technical Survey," Working Papers halshs-01145485, HAL.
    20. Driver, Ciaran & Temple, Paul & Urga, Giovanni, 2008. "Real options -- delay vs. pre-emption: Do industrial characteristics matter?," International Journal of Industrial Organization, Elsevier, vol. 26(2), pages 532-545, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2346-:d:1149599. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.