A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equations

TL;DR

This paper introduces a deep learning method for solving path-dependent PDEs that overcomes basis selection limitations, providing accurate approximations with proven error bounds and demonstrating superior performance in high-dimensional financial applications.

Contribution

It presents a novel deep learning framework for PPDEs that eliminates the need for basis functions and establishes error bounds, improving accuracy especially in high-dimensional problems.

Findings

More accurate than existing deep learning methods

Effective in high-dimensional Asian and barrier options

Provides theoretical error bounds for approximations

Abstract

Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using regression. However, a limitation of this approach is to require the selection of a basis in a function space. In this paper, we overcome this limitation by the use of deep learning methods, and we show that this setting allows for the derivation of error bounds on the approximation of conditional expectations. Numerical examples based on a two-person zero-sum game, as well as on Asian and barrier option pricing, are presented. In comparison with other deep learning approaches, our algorithm appears to be more accurate, especially in large dimensions.