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

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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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