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

Akihiko Takahashi, Yoshifumi Tsuchida and Toshihiro Yamada
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Akihiko Takahashi: The University of Tokyo, Tokyo, Japan
Yoshifumi Tsuchida: Hitotsubashi University, Tokyo, Japan
Toshihiro Yamada: Hitotsubashi University, Tokyo, Japan, Japan Science and Technology Agency (JST), Tokyo, Japan

No CARF-F-504, CARF F-Series from Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo

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, linear PDE part and nonlinear PDE part. 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.

Pages: 38
Date: 2021-01
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