  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.
