Abstract
The paper develops a new deep learning based scheme for solving high-dimensional nonlinear forward-backward stochastic differential equations (FBSDE) and associated partial differential equations. Firstly, the original BSDE is split into the linear dominant BSDE part and the nonlinear residual BSDE part. Then the linear BSDE part is approximated with high accuracy using a weak approximation technique. To approximate the nonlinear BSDE part, Deep BSDE solver is applied with asymptotic expansions which work as control variates. A sharp error estimate provides how the new scheme improves the original Deep BSDE method. Numerical experiments for high-dimensional nonlinear models show the validity and the effectiveness of the new scheme in financial application.
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Acknowledgements
I would like to express my sincere gratitude to Prof. Akihiko Takahashi (University of Tokyo) and Prof. Toshihiro Yamada (Hitotsubashi University) for their comments and advice on the method of this paper. Also, I would like to thank Mr. Riu Naito (Japan Post Insurance Co., Ltd. & Hitotsubashi University) for his support on numerical experiments. Lastly, I thank two anonymous reviewers for their grateful comments and suggestions.
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Tsuchida, Y. Control Variate Method for Deep BSDE Solver Using Weak Approximation. Asia-Pac Financ Markets 30, 273–296 (2023). https://doi.org/10.1007/s10690-022-09374-8
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DOI: https://doi.org/10.1007/s10690-022-09374-8