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A new efficient approximation scheme for solving high-dimensional semilinear PDEs: control variate method for Deep BSDE solver

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  • Akihiko Takahashi
  • Yoshifumi Tsuchida
  • Toshihiro Yamada

Abstract

This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into two parts, namely "dominant" linear and "small" nonlinear PDEs. Then, we employ a Deep BSDE solver with a new control variate method to solve those PDEs, where approximations based on an asymptotic expansion technique are effectively applied to the linear part and also used as control variates for the nonlinear part. Moreover, our theoretical result indicates that errors of the proposed method become much smaller than those of the original Deep BSDE solver. Finally, we show numerical experiments to demonstrate the validity of our method, which is consistent with the theoretical result in this paper.

Suggested Citation

  • 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," Papers 2101.09890, arXiv.org, revised Jan 2021.
  • Handle: RePEc:arx:papers:2101.09890
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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, 2015. "An Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver," CARF F-Series CARF-F-363, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Riu Naito & Toshihiro Yamada, 2020. "An acceleration scheme for deep learning-based BSDE solver using weak expansions," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 7(02), pages 1-12, June.
    4. Masaaki Fujii & Akihiko Takahashi, 2011. "Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme," CIRJE F-Series CIRJE-F-802, CIRJE, Faculty of Economics, University of Tokyo.
    5. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method (Published in "Journal of Japan Statistical Society", Vol.35-2, 171-203, 2005. )," CARF F-Series CARF-F-030, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    6. Akihiko Takahashi, 2015. "Asymptotic Expansion Approach in Finance," CARF F-Series CARF-F-356, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2015.
    7. Alessandro Gnoatto & Athena Picarelli & Christoph Reisinger, 2020. "Deep xVA solver - A neural network based counterparty credit risk management framework," Working Papers 07/2020, University of Verona, Department of Economics.
    8. Akihiko Takahashi & Toshihiro Yamada, 2015. "An asymptotic expansion of forward-backward SDEs with a perturbed driver," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-29.
    9. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    10. 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.
    11. 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.
    12. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs," CARF F-Series CARF-F-372, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    13. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    14. Akihiko Takahashi & Toshiaki Watanabe, 2015. "An Asymptotic Expansion of Forward-Backward SDEs with a Perturbed Driver ," CIRJE F-Series CIRJE-F-976, CIRJE, Faculty of Economics, University of Tokyo.
    15. Masaaki Fujii & Akihiko Takahashi, 2011. "Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme," CARF F-Series CARF-F-248, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    16. 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.
    17. Fujii, Masaaki & Takahashi, Akihiko, 2019. "Solving backward stochastic differential equations with quadratic-growth drivers by connecting the short-term expansions," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1492-1532.
    18. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," CIRJE F-Series CIRJE-F-993, CIRJE, Faculty of Economics, University of Tokyo.
    19. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method," CIRJE F-Series CIRJE-F-335, CIRJE, Faculty of Economics, University of Tokyo.
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    Cited by:

    1. Yuga Iguchi & Riu Naito & Yusuke Okano & Akihiko Takahashi & Toshihiro Yamada, 2021. "Deep Asymptotic Expansion: Application to Financial Mathematics," CIRJE F-Series CIRJE-F-1178, CIRJE, Faculty of Economics, University of Tokyo.
    2. Yuga Iguchi & Riu Naito & Yusuke Okano & Akihiko Takahashi & Toshihiro Yamada, 2021. "Deep Asymptotic Expansion with Weak Approximation ," CIRJE F-Series CIRJE-F-1168, CIRJE, Faculty of Economics, University of Tokyo.
    3. Jean-Franc{c}ois Chassagneux & Junchao Chen & Noufel Frikha, 2022. "Deep Runge-Kutta schemes for BSDEs," Papers 2212.14372, arXiv.org.
    4. Yuga Iguchi & Riu Naito & Yusuke Okano & Akihiko Takahashi & Toshihiro Yamada, 2021. "Deep Asymptotic Expansion: Application to Financial Mathematics(forthcoming in proceedings of IEEE CSDE 2021)," CARF F-Series CARF-F-523, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.

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