Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning

TL;DR

This paper explores the application of the Deep Galerkin Method, a mesh-free deep learning approach, to solve high-dimensional PDEs in finance, highlighting its advantages over traditional numerical methods.

Contribution

It demonstrates the effectiveness of DGM in solving complex PDEs in finance, analyzing its features, capabilities, and limitations across various applications.

Findings

DGM effectively solves high-dimensional PDEs in finance.

The method reduces computational complexity compared to traditional approaches.

DGM shows promise in handling PDE systems with fewer discretization constraints.

Abstract

In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal execution, mean field games, etc. The main idea behind DGM is to represent the unknown function of interest using a deep neural network. A key feature of this approach is the fact that, unlike other commonly used numerical approaches such as finite difference methods, it is mesh-free. As such, it does not suffer (as much as other numerical methods) from the curse of dimensionality associated with highdimensional PDEs and PDE systems. The main goals of this paper are to elucidate the features, capabilities and limitations of DGM by analyzing aspects of its implementation for a number of different PDEs and PDE systems. Additionally, we present: (1) a brief overview of PDEs in quantitative finance along with numerical methods for solving them; (2) a brief overview of deep learning and, in particular, the notion of neural networks; (3) a discussion of the theoretical foundations of DGM with a focus on the justification of why this method is expected to perform well.