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

TL;DR

This paper presents a novel approximation scheme combining decomposition and control variates with Deep BSDE to efficiently solve high-dimensional semilinear PDEs, significantly reducing errors.

Contribution

It introduces a decomposition-based control variate method for Deep BSDE, improving accuracy for high-dimensional semilinear PDEs using asymptotic expansion techniques.

Findings

Errors are significantly reduced compared to original Deep BSDE.

Numerical experiments validate the theoretical error reduction.

Method effectively handles high-dimensional problems.

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 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.