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Solving Kolmogorov PDEs without the curse of dimensionality via deep learning and asymptotic expansion with Malliavin calculus

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  • Akihiko Takahashi

    (The University of Tokyo)

  • Toshihiro Yamada

    (Hitotsubashi University
    Japan Science and Technology Agency (JST))

Abstract

This paper proposes a new spatial approximation method without the curse of dimensionality for solving high-dimensional partial differential equations (PDEs) by using an asymptotic expansion method with a deep learning-based algorithm. In particular, the mathematical justification on the spatial approximation is provided. Numerical examples for high-dimensional Kolmogorov PDEs show effectiveness of our method.

Suggested Citation

  • Akihiko Takahashi & Toshihiro Yamada, 2023. "Solving Kolmogorov PDEs without the curse of dimensionality via deep learning and asymptotic expansion with Malliavin calculus," Partial Differential Equations and Applications, Springer, vol. 4(4), pages 1-31, August.
    • Handle: RePEc:spr:pardea:v:4:y:2023:i:4:d:10.1007_s42985-023-00240-4
    DOI: 10.1007/s42985-023-00240-4
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    References listed on IDEAS

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    1. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," Papers 1710.07030, arXiv.org, revised Mar 2019.
    2. Akihiko Takahashi & Toshihiro Yamada, 2012. "A Remark on Approximation of the Solutions to Partial Differential Equations in Finance," CARF F-Series CARF-F-273, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Mar 2012.
    3. Akihiko Takahashi & Toshihiro Yamada, 2012. "On Approximation of the Solutions to Partial Differential Equations in Finance," CARF F-Series CARF-F-249, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Mar 2012.
    4. Akihiko Takahashi & Toshihiro Yamada, 2012. "A Remark on Approximation of the Solutions to Partial Differential Equations in Finance," CIRJE F-Series CIRJE-F-842, CIRJE, Faculty of Economics, University of Tokyo.
    5. Naoto Kunitomo & Akihiko Takahashi, 2001. "The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 117-151, January.
    6. Akihiko Takahashi & Toshihiro Yamada, 2012. "A Remark on Approximation of the Solutions to Partial Differential Equations in Finance," World Scientific Book Chapters, in: Akihiko Takahashi & Yukio Muromachi & Hidetaka Nakaoka (ed.), Recent Advances In Financial Engineering 2011, chapter 8, pages 133-181, World Scientific Publishing Co. Pte. Ltd..
    7. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 26(3), pages 391-408, September.
    8. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    9. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method," CIRJE F-Series CIRJE-F-335, CIRJE, Faculty of Economics, University of Tokyo.
    10. Akihiko Takahashi & Yoshifumi Tsuchida & Toshihiro Yamada, 2021. "A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver," CARF F-Series CARF-F-504, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2022.
    11. Akihiko Takahashi & Toshihiro Yamada, 2011. "On Approximation of the Solutions to Partial Differential Equations in Finance," CIRJE F-Series CIRJE-F-815, CIRJE, Faculty of Economics, University of Tokyo.
    12. Akihiko Takahashi & Toshihiro Yamada, 2015. "On Error Estimates for Asymptotic Expansions with Malliavin Weights: Application to Stochastic Volatility Model," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 513-541, March.
    13. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs (Forthcoming in Asia-Pacific Financial Markets)," CARF F-Series CARF-F-456, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
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    Cited by:

    1. Lukas Gonon, 2024. "Deep neural network expressivity for optimal stopping problems," Finance and Stochastics, Springer, vol. 28(3), pages 865-910, July.

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