A forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations

TL;DR

This paper introduces a novel deep learning algorithm that efficiently solves high-dimensional nonlinear backward stochastic differential equations by transforming the problem into a differential deep learning framework using Malliavin calculus.

Contribution

The work develops a forward differential deep learning method that approximates solutions, gradients, and Hessians of BSDEs, improving accuracy and efficiency over existing methods.

Findings

More accurate solutions in high dimensions

Reduced computation time compared to existing methods

Effective handling of nonlinear BSDEs with neural networks

Abstract

In this work, we present a novel forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs). Motivated by the fact that differential deep learning can efficiently approximate the labels and their derivatives with respect to inputs, we transform the BSDE problem into a differential deep learning problem. This is done by leveraging Malliavin calculus, resulting in a system of BSDEs. The unknown solution of the BSDE system is a triple of processes $(Y, Z, \Gamma)$, representing the solution, its gradient, and the Hessian matrix. The main idea of our algorithm is to discretize the integrals using the Euler-Maruyama method and approximate the unknown discrete solution triple using three deep neural networks. The parameters of these networks are then optimized by globally minimizing a differential learning loss function, which is novelty defined as a weighted sum of the dynamics of the discretized system of BSDEs. Through various high-dimensional examples, we demonstrate that our proposed scheme is more efficient in terms of accuracy and computation time compared to other contemporary forward deep learning-based methodologies.