A New Efficient Approximation Scheme for Solving High-Dimensional Semilinear PDEs: Control Variate Method for Deep BSDE Solver

29 Pages Posted: 22 Jan 2021

See all articles by Akihiko Takahashi

Akihiko Takahashi

University of Tokyo - Faculty of Economics

Yoshifumi Tsuchida

Hitotsubashi University

Toshihiro Yamada

Hitotsubashi University

Date Written: January 21, 2021

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, namely "dominant" linear and "small" nonlinear PDEs. Then, we apply 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.

Keywords: Deep learning, Semilinear partial differential equations, Backward stochastic differential equations, Deep BSDE solver, Asymptotic expansion, Control variate method

JEL Classification: C15, C61, G11, G13

Suggested Citation

Takahashi, Akihiko and Tsuchida, Yoshifumi and Yamada, Toshihiro, A New Efficient Approximation Scheme for Solving High-Dimensional Semilinear PDEs: Control Variate Method for Deep BSDE Solver (January 21, 2021). Available at SSRN: https://ssrn.com/abstract=3770397 or http://dx.doi.org/10.2139/ssrn.3770397

Akihiko Takahashi (Contact Author)

University of Tokyo - Faculty of Economics ( email )

7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033
Japan

Yoshifumi Tsuchida

Hitotsubashi University ( email )

2-1 Naka Kunitachi-shi
Tokyo 186-8601
Japan

Toshihiro Yamada

Hitotsubashi University ( email )

2-1 Naka Kunitachi-shi
Tokyo 186-8601
Japan

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
18
Abstract Views
121
PlumX Metrics