IDEAS home Printed from https://ideas.repec.org/a/spr/pardea/v5y2024i6d10.1007_s42985-024-00272-4.html
   My bibliography  Save this article

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Author

Listed:
  • Christian Beck

    (ETH Zurich)

  • Lukas Gonon

    (Imperial College London)

  • Arnulf Jentzen

    (The Chinese University of Hong Kong (CUHK-Shenzhen)
    University of Münster)

Abstract

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs.

Suggested Citation

  • Christian Beck & Lukas Gonon & Arnulf Jentzen, 2024. "Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations," Partial Differential Equations and Applications, Springer, vol. 5(6), pages 1-47, December.
    • Handle: RePEc:spr:pardea:v:5:y:2024:i:6:d:10.1007_s42985-024-00272-4
    DOI: 10.1007/s42985-024-00272-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s42985-024-00272-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s42985-024-00272-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Philipp Grohs & Fabian Hornung & Arnulf Jentzen & Philippe von Wurstemberger, 2018. "A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations," Papers 1809.02362, arXiv.org, revised Jan 2023.
    2. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2014. "A numerical algorithm for a class of BSDEs via the branching process," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1112-1140.
    3. Ludovic Gouden`ege & Andrea Molent & Antonino Zanette, 2019. "Machine Learning for Pricing American Options in High-Dimensional Markovian and non-Markovian models," Papers 1905.09474, arXiv.org, revised Jun 2019.
    4. Weinan E & Martin Hutzenthaler & Arnulf Jentzen & Thomas Kruse, 2021. "Multilevel Picard iterations for solving smooth semilinear parabolic heat equations," Partial Differential Equations and Applications, Springer, vol. 2(6), pages 1-31, December.
    5. Yangang Chen & Justin W. L. Wan, 2021. "Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions," Quantitative Finance, Taylor & Francis Journals, vol. 21(1), pages 45-67, January.
    6. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CIRJE F-Series CIRJE-F-1069, CIRJE, Faculty of Economics, University of Tokyo.
    7. Sebastian Becker & Patrick Cheridito & Arnulf Jentzen & Timo Welti, 2019. "Solving high-dimensional optimal stopping problems using deep learning," Papers 1908.01602, arXiv.org, revised Aug 2021.
    8. Agarwal, Ankush & Claisse, Julien, 2020. "Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5006-5036.
    9. Rasulov, A. & Raimova, G. & Mascagni, M., 2010. "Monte Carlo solution of Cauchy problem for a nonlinear parabolic equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(6), pages 1118-1123.
    10. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    11. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," CARF F-Series CARF-F-423, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    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. Andrew Na & Justin Wan, 2023. "Efficient Pricing and Hedging of High Dimensional American Options Using Recurrent Networks," Papers 2301.08232, arXiv.org.
    2. Martin Hutzenthaler & Arnulf Jentzen & Thomas Kruse & Tuan Anh Nguyen, 2020. "A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-34, April.
    3. Lukas Gonon, 2024. "Deep neural network expressivity for optimal stopping problems," Finance and Stochastics, Springer, vol. 28(3), pages 865-910, July.
    4. Philipp Grohs & Arnulf Jentzen & Diyora Salimova, 2022. "Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms," Partial Differential Equations and Applications, Springer, vol. 3(4), pages 1-41, August.
    5. Lukas Gonon, 2022. "Deep neural network expressivity for optimal stopping problems," Papers 2210.10443, arXiv.org.
    6. Stefan Kremsner & Alexander Steinicke & Michaela Szölgyenyi, 2020. "A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics," Risks, MDPI, vol. 8(4), pages 1-18, December.
    7. Ariel Neufeld & Philipp Schmocker & Sizhou Wu, 2024. "Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs," Papers 2405.05192, arXiv.org, revised Jan 2025.
    8. Stefan Kremsner & Alexander Steinicke & Michaela Szolgyenyi, 2020. "A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics," Papers 2010.15757, arXiv.org, revised Dec 2020.
    9. Yangang Chen & Justin W. L. Wan, 2019. "Deep Neural Network Framework Based on Backward Stochastic Differential Equations for Pricing and Hedging American Options in High Dimensions," Papers 1909.11532, arXiv.org.
    10. Masafumi Nakano & Akihiko Takahashi & Soichiro Takahashi, 2018. "Bitcoin technical trading with artificial neural network," CIRJE F-Series CIRJE-F-1078, CIRJE, Faculty of Economics, University of Tokyo.
    11. Masafumi Nakano & Akihiko Takahashi & Soichiro Takahashi, 2018. "Bitcoin technical trading with artificial neural network," CARF F-Series CARF-F-430, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    12. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    13. Sebastian Becker & Patrick Cheridito & Arnulf Jentzen, 2020. "Pricing and Hedging American-Style Options with Deep Learning," JRFM, MDPI, vol. 13(7), pages 1-12, July.
    14. Agarwal, Ankush & Claisse, Julien, 2020. "Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5006-5036.
    15. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2023. "Deep stochastic optimization in finance," Digital Finance, Springer, vol. 5(1), pages 91-111, March.
    16. Beatriz Salvador & Cornelis W. Oosterlee & Remco van der Meer, 2020. "Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks," Mathematics, MDPI, vol. 9(1), pages 1-20, December.
    17. Kathrin Glau & Linus Wunderlich, 2020. "The Deep Parametric PDE Method: Application to Option Pricing," Papers 2012.06211, arXiv.org.
    18. Chinonso Nwankwo & Nneka Umeorah & Tony Ware & Weizhong Dai, 2022. "Deep learning and American options via free boundary framework," Papers 2211.11803, arXiv.org, revised Dec 2022.
    19. Nakano, Masafumi & Takahashi, Akihiko & Takahashi, Soichiro, 2018. "Bitcoin technical trading with artificial neural network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 510(C), pages 587-609.
    20. Laurens Van Mieghem & Antonis Papapantoleon & Jonas Papazoglou-Hennig, 2023. "Machine learning for option pricing: an empirical investigation of network architectures," Papers 2307.07657, arXiv.org.

    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:spr:pardea:v:5:y:2024:i:6:d:10.1007_s42985-024-00272-4. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.