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

Asia-Pacific Financial Markets volume 26, pages 391–408 (2019)Cite this article

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 et al. (Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, 2017b. arXiv:1706.04702), drastically reduces the loss function and accelerates the speed of convergence. We illustrate the technique and its implications by using Bergman’s model with different lending and borrowing rates as a typical model for FVA as well as a class of solvable BSDEs with quadratic growth drivers. We also present an extension of the deep BSDE solver for reflected BSDEs representing American option prices.

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Notes

  1. See also Beck et al. (2017), where the method is applied to different types of BSDEs, and Weinan et al. (2016, 2017a) as different approaches to high-dimensional problems.

  2. As interesting applications of machine learning to various investment strategies, see Nakano et al. (2017a, b, c).

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Acknowledgements

The research is partially supported by Center for Advanced Research in Finance (CARF).

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Authors and Affiliations

  1. Quantitative Finance Course, Graduate School of Economics, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Masaaki Fujii, Akihiko Takahashi & Masayuki Takahashi

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  1. Masaaki Fujii
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  2. Akihiko Takahashi
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  3. Masayuki Takahashi
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Correspondence to Akihiko Takahashi.

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Fujii, M., Takahashi, A. & Takahashi, M. Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs. Asia-Pac Financ Markets 26, 391–408 (2019). https://doi.org/10.1007/s10690-019-09271-7

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  • DOI: https://doi.org/10.1007/s10690-019-09271-7

Keywords

  • Deep learning
  • BSDEs
  • Asymptotic expansion
  • Deep BSDE solver
  • FVA
  • American option
  • High dimensional BSDEs
  • Different lending
  • Borrowing rates