IDEAS home Printed from https://ideas.repec.org/p/aiz/louvad/2025012.html

Optimal control by policy improvements and constrained Gaussian process regressions

Author

Listed:
  • Hainaut, Donatien

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Dupret, Jean-Loup

    (ETH Zürich)

Abstract

This article proposes a new iterative algorithm combining policy improvements and Gaussian process regressions for solving stochastic control problems. At each iteration, we update an approximated value function and then improve the controls. We sample state variables in the inner domain and on the terminal boundary of the Hamilton-Jacobi-Bellman (HJB) equation. The approximated value function, the solution of this equation, is obtained by fitting a constrained regression function. The regression function matches the terminal utility on the boundary sample and satisfies the HJB equation on the inner sample. Assuming the regression function is a Gaussian process, we find a closed-form approximation of the value function and of the controls. In a numerical illustration, we test the efficiency of the method for solving the consumption-investment and linear-quadratic regulator problems.

Suggested Citation

  • Hainaut, Donatien & Dupret, Jean-Loup, 2025. "Optimal control by policy improvements and constrained Gaussian process regressions," LIDAM Discussion Papers ISBA 2025012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2025012
    as

    Download full text from publisher

    File URL: https://dial.uclouvain.be/pr/boreal/fr/object/boreal%3A301229/datastream/PDF_01/view
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Price, Ilan & Fowkes, Jaroslav & Hopman, Daniel, 2019. "Gaussian processes for unconstraining demand," European Journal of Operational Research, Elsevier, vol. 275(2), pages 621-634.
    2. Charles H. Fine & Evan L. Porteus, 1989. "Dynamic Process Improvement," Operations Research, INFORMS, vol. 37(4), pages 580-591, August.
    3. Hainaut, Donatien, 2024. "American option pricing with model constrained Gaussian process regressions," LIDAM Discussion Papers ISBA 2024023, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Kathrin Glau & Linus Wunderlich, 2024. "Neural network expression rates and applications of the deep parametric PDE method in counterparty credit risk," Annals of Operations Research, Springer, vol. 336(1), pages 331-357, May.
    5. Hainaut, Donatien & Casas, Alex, 2024. "Option pricing in the Heston model with physics inspired neural networks," LIDAM Reprints ISBA 2024043, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. Hainaut, Donatien & Casas, Alex, 2024. "Option pricing in the Heston model with Physics inspired neural networks," LIDAM Discussion Papers ISBA 2024002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Jan De Spiegeleer & Dilip B. Madan & Sofie Reyners & Wim Schoutens, 2018. "Machine learning for quantitative finance: fast derivative pricing, hedging and fitting," Quantitative Finance, Taylor & Francis Journals, vol. 18(10), pages 1635-1643, October.
    8. Donatien Hainaut & Alex Casas, 2024. "Option pricing in the Heston model with physics inspired neural networks," Annals of Finance, Springer, vol. 20(3), pages 353-376, September.
    9. Dupret, Jean-Loup & Hainaut, Donatien, 2024. "Deep learning for high-dimensional continuous-time stochastic optimal control without explicit solution," LIDAM Discussion Papers ISBA 2024016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Hainaut, Donatien & Vrins, Frédéric, 2024. "European option pricing with model constrained Gaussian process regressions," LIDAM Discussion Papers LFIN 2024005, Université catholique de Louvain, Louvain Finance (LFIN).
    11. Glau, Kathrin & Wunderlich, Linus, 2022. "The deep parametric PDE method and applications to option pricing," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    12. Achref Bachouch & Côme Huré & Nicolas Langrené & Huyên Pham, 2022. "Deep Neural Networks Algorithms for Stochastic Control Problems on Finite Horizon: Numerical Applications," Methodology and Computing in Applied Probability, Springer, vol. 24(1), pages 143-178, March.
    13. Al-Aradi, Ali & Correia, Adolfo & Jardim, Gabriel & de Freitas Naiff, Danilo & Saporito, Yuri, 2022. "Extensions of the deep Galerkin method," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    14. Jeongsub Choi & Youngdoo Son & Myong K. Jeong, 2024. "Gaussian kernel with correlated variables for incomplete data," Annals of Operations Research, Springer, vol. 341(1), pages 223-244, October.
    15. Ali Al-Aradi & Adolfo Correia & Danilo de Frietas Naiff & Gabriel Jardim & Yuri Saporito, 2019. "Extensions of the Deep Galerkin Method," Papers 1912.01455, arXiv.org, revised Apr 2022.
    16. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hainaut, Donatien, 2026. "American option pricing with model constrained Gaussian process regressions," Applied Mathematics and Computation, Elsevier, vol. 512(C).

    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. Hainaut, Donatien, 2026. "American option pricing with model constrained Gaussian process regressions," Applied Mathematics and Computation, Elsevier, vol. 512(C).
    2. Hainaut, Donatien, 2024. "American option pricing with model constrained Gaussian process regressions," LIDAM Discussion Papers ISBA 2024023, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. Tan, Jianguo & Zhang, Xingyu, 2026. "Improved constrained physics-informed neural networks (ICPINNs) to solve PDE and its application to option pricing," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PA), pages 908-924.
    4. Hainaut, Donatien & Casas, Alex, 2024. "Option pricing in the Heston model with Physics inspired neural networks," LIDAM Discussion Papers ISBA 2024002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Lijie Ding & Egang Lu & Kin Cheung, 2025. "Deep Learning Option Pricing with Market Implied Volatility Surfaces," Papers 2509.05911, arXiv.org.
    6. Naman Krishna Pande & Puneet Pasricha & Arun Kumar & Arvind Kumar Gupta, 2024. "European Option Pricing in Regime Switching Framework via Physics-Informed Residual Learning," Papers 2410.10474, arXiv.org.
    7. Victor Boussange & Sebastian Becker & Arnulf Jentzen & Benno Kuckuck & Loïc Pellissier, 2023. "Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions," Partial Differential Equations and Applications, Springer, vol. 4(6), pages 1-51, December.
    8. Choi, So Eun & Jang, Hyun Jin & Lee, Kyungsub & Zheng, Harry, 2021. "Optimal market-Making strategies under synchronised order arrivals with deep neural networks," Journal of Economic Dynamics and Control, Elsevier, vol. 125(C).
    9. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    10. Wenxuan Zhang & Yixiao Guo & Benzhuo Lu, 2025. "Exact Terminal Condition Neural Network for American Option Pricing Based on the Black-Scholes-Merton Equations," Papers 2510.27132, arXiv.org.
    11. Donatien Hainaut & Alex Casas, 2024. "Option pricing in the Heston model with physics inspired neural networks," Annals of Finance, Springer, vol. 20(3), pages 353-376, September.
    12. Gero Junike & Hauke Stier, 2024. "Enhancing Fourier pricing with machine learning," Papers 2412.05070, arXiv.org.
    13. Edson Pindza & Jules Clement & Sutene Mwambi & Nneka Umeorah, 2025. "Neural Network for Valuing Bitcoin Options Under Jump-Diffusion and Market Sentiment Model," Computational Economics, Springer;Society for Computational Economics, vol. 66(3), pages 2305-2342, September.
    14. Oumar Mbodji & Traian A. Pirvu, 2025. "Portfolio time consistency and utility weighted discount rates," Mathematics and Financial Economics, Springer, volume 19, number 2, January.
    15. Dupret, Jean-Loup & Hainaut, Donatien, 2024. "Deep learning for high-dimensional continuous-time stochastic optimal control without explicit solution," LIDAM Discussion Papers ISBA 2024016, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Rushikesh Handal & Masanori Hirano, 2026. "KANHedge: Efficient Hedging of High-Dimensional Options Using Kolmogorov-Arnold Network-Based BSDE Solver," Papers 2601.11097, arXiv.org.
    17. Olivier Bokanowski & Averil Prost & Xavier Warin, 2023. "Neural networks for first order HJB equations and application to front propagation with obstacle terms," Partial Differential Equations and Applications, Springer, vol. 4(5), pages 1-36, October.
    18. Chang, Shuhua & Hu, Kaixuan & Wang, Xinyu, 2025. "Optimal carbon emission path under uncertainty: Physical risks and transition risks," Energy Economics, Elsevier, vol. 151(C).
    19. Kathrin Glau & Linus Wunderlich, 2024. "Neural network expression rates and applications of the deep parametric PDE method in counterparty credit risk," Annals of Operations Research, Springer, vol. 336(1), pages 331-357, May.
    20. Hainaut, Donatien & Vrins, Frédéric, 2024. "European option pricing with model constrained Gaussian process regressions," LIDAM Discussion Papers LFIN 2024005, Université catholique de Louvain, Louvain Finance (LFIN).

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:aiz:louvad:2025012. 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: Nadja Peiffer (email available below). General contact details of provider: https://edirc.repec.org/data/isuclbe.html .

    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.