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

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