Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations

TL;DR

This paper introduces a novel deep learning approach for efficiently solving high-dimensional nonlinear backward stochastic differential equations by reformulating the problem as a global optimization task.

Contribution

It presents a new deep neural network-based scheme that approximates solutions of BSDEs through global minimization of local loss functions, improving convergence and accuracy.

Findings

Effective in high-dimensional settings

Demonstrated on finance pricing problems

Achieves accurate solutions with neural networks

Abstract

In this work, we propose a new deep learning-based scheme for solving high dimensional nonlinear backward stochastic differential equations (BSDEs). The idea is to reformulate the problem as a global optimization, where the local loss functions are included. Essentially, we approximate the unknown solution of a BSDE using a deep neural network and its gradient with automatic differentiation. The approximations are performed by globally minimizing the quadratic local loss function defined at each time step, which always includes the terminal condition. This kind of loss functions are obtained by iterating the Euler discretization of the time integrals with the terminal condition. Our formulation can prompt the stochastic gradient descent algorithm not only to take the accuracy at each time layer into account, but also converge to a good local minima. In order to demonstrate performances of our algorithm, several high-dimensional nonlinear BSDEs including pricing problems in finance are provided.