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Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs

Author

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  • Masaaki Fujii

    (Quantitative Finance Course, Graduate School of Economics, The University of Tokyo)

  • Akihiko Takahashi

    (Quantitative Finance Course, Graduate School of Economics, The University of Tokyo)

  • Masayuki Takahashi

    (Quantitative Finance Course, Graduate School of Economics, The University of Tokyo)

Abstract

We demonstrate that the use of asymptotic expansion as prior knowledge in the "deep BSDE solver", which is a deep learning method for high dimensional BSDEs proposed by Weinan E, Han & Jentzen (2017), drastically reduces the loss function and accelerates the speed of convergence. We illustrate the technique and its implications by Bergman's model with different lending and borrowing rates, and a class of quadratic-growth BSDEs. We also present an extension of the deep BSDE solver for reflected BSDEs using an American basket option as an example.

Suggested Citation

  • 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.
    • Handle: RePEc:cfi:fseres:cf423
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    References listed on IDEAS

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    5. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations with quadratic-growth drivers by Connecting the Short-term Expansions," Papers 1606.04285, arXiv.org, revised May 2018.
    6. 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.
    7. Masafumi Nakano & Akihiko Takahashi & Soichiro Takahashi, 2017. "Robust Technical Trading with Fuzzy Knowledge-based Systems," CIRJE F-Series CIRJE-F-1053, CIRJE, Faculty of Economics, University of Tokyo.
    8. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions(Revised version of CARF-F-387)," CARF F-Series CARF-F-398, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
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    16. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility," CIRJE F-Series CIRJE-F-840, CIRJE, Faculty of Economics, University of Tokyo.
    17. Masafumi Nakano & Akihiko Takahashi & Soichiro Takahashi, 2017. "Fuzzy Logic-based Portfolio Selection with Particle Filtering and Anomaly Detection," CIRJE F-Series CIRJE-F-1037, CIRJE, Faculty of Economics, University of Tokyo.
    18. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions," CARF F-Series CARF-F-387, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    19. 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.
    20. Masaaki Fujii & Akihiko Takahashi, 2011. "Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme," Papers 1106.0123, arXiv.org, revised Jan 2012.
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    22. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility," Papers 1202.0608, arXiv.org, revised Sep 2012.
    23. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    24. Masaaki Fujii & Akihiko Takahashi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(3), pages 283-304, September.
    25. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," Papers 1510.03220, arXiv.org, revised Sep 2018.
    26. 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.
    27. Stéphane Crépey & Shiqi Song, 2016. "Counterparty risk and funding: immersion and beyond," Finance and Stochastics, Springer, vol. 20(4), pages 901-930, October.
    28. Masafumi Nakano & Akihiko Takahashi & Muhammad Soichiro Takahashi, 2017. "Creating Investment Scheme with State Space Modeling," CIRJE F-Series CIRJE-F-1038, CIRJE, Faculty of Economics, University of Tokyo.
    29. Stéphane Crépey & Shiqi Song, 2017. "Invariance properties in the dynamic gaussian copula model ," Working Papers hal-01455424, HAL.
    30. 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.
    31. Masaaki Fujii & Akihiko Takahashi, 2016. "Solving Backward Stochastic Differential Equations by Connecting the Short-term Expansions," CIRJE F-Series CIRJE-F-1016, CIRJE, Faculty of Economics, University of Tokyo.
    32. Masaaki Fujii & Akihiko Takahashi, 2012. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CARF F-Series CARF-F-278, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2015.
    33. Masaaki Fujii, Akihiko Takahashi, 2012. "Analytical Approximation for Non-linear FBSDEs with Perturbation Scheme," CARF F-Series CARF-F-269, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    34. Masaaki Fujii & Akihiko Takahshi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CIRJE F-Series CIRJE-F-954, CIRJE, Faculty of Economics, University of Tokyo.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. Masafumi Nakano & Akihiko Takahashi & Soichiro Takahashi, 2018. "Bitcoin Technical Trading with Articial Neural Network," CIRJE F-Series CIRJE-F-1090, CIRJE, Faculty of Economics, University of Tokyo.
    6. Andrew Na & Justin Wan, 2023. "Efficient Pricing and Hedging of High Dimensional American Options Using Recurrent Networks," Papers 2301.08232, arXiv.org.
    7. 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.
    8. 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.
    9. 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.
    10. Warin Xavier, 2018. "Nesting Monte Carlo for high-dimensional non-linear PDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 24(4), pages 225-247, December.

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