Gradient boosting-based numerical methods for high-dimensional backward stochastic differential equations

TL;DR

This paper introduces a novel algorithm combining theta-discretization and XGBoost regression to efficiently solve high-dimensional nonlinear backward stochastic differential equations, demonstrating high accuracy up to 10,000 dimensions.

Contribution

The paper presents a new method integrating theta-discretization with XGBoost for high-dimensional BSDEs, enabling efficient and accurate solutions in very high dimensions.

Findings

Successfully solves BSDEs up to 10,000 dimensions

Demonstrates high efficiency and accuracy of the proposed method

Outperforms existing approaches in high-dimensional settings

Abstract

In this work we propose a new algorithm for solving high-dimensional backward stochastic differential equations (BSDEs). Based on the general theta-discretization for the time-integrands, we show how to efficiently use eXtreme Gradient Boosting (XGBoost) regression to approximate the resulting conditional expectations in a quite high dimension. Numerical results illustrate the efficiency and accuracy of our proposed algorithms for solving very high-dimensional (up to $10000$ dimensions) nonlinear BSDEs.