Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations

TL;DR

This paper introduces a new Monte Carlo-based numerical method for high-dimensional nonlinear BSDEs that overcomes the curse of dimensionality, enabling efficient approximation regardless of the problem's dimension.

Contribution

The paper presents the first proven Monte Carlo-type method that overcomes the curse of dimensionality in approximating solutions to nonlinear high-dimensional BSDEs.

Findings

The new method's computational complexity grows polynomially with inverse accuracy and dimension.

The method is proven to overcome the curse of dimensionality for nonlinear BSDEs.

Numerical experiments validate the theoretical results.

Abstract

Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical approximation methods to approximatively solve nonlinear high-dimensional BSDEs. Although there are a large number of research articles in the scientific literature which analyze numerical approximation methods for nonlinear BSDEs, until today there has been no numerical approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of nonlinear BSDEs in the sense that the number of computational operations of the numerical approximation method to approximatively compute one sample path of the BSDE solution grows at most polynomially in both the reciprocal $1/ \varepsilon$ of the prescribed approximation accuracy $\varepsilon \in (0,\infty)$ and the dimension $d\in \mathbb N=\{1,2,3,\ldots\}$ of the BSDE. It is the key contribution of this article to overcome this obstacle by introducing a new Monte Carlo-type numerical approximation method for high-dimensional BSDEs and by proving that this Monte Carlo-type numerical approximation method does indeed overcome the curse of dimensionality in the approximative computation of solution paths of BSDEs.