  Applications in quantitative finance such as optimal trade execution, risk
management of options, and optimal asset allocation involve the solution of
high dimensional and nonlinear Partial Differential Equations (PDEs). The
connection between PDEs and systems of Forward-Backward Stochastic Differential
Equations (FBSDEs) enables the use of advanced simulation techniques to be
applied even in the high dimensional setting. Unfortunately, when the
underlying application contains nonlinear terms, then classical methods both
for simulation and numerical methods for PDEs suffer from the curse of
dimensionality. Inspired by the success of deep learning, several researchers
have recently proposed to address the solution of FBSDEs using deep learning.
We discuss the dynamical systems point of view of deep learning and compare
several architectures in terms of stability, generalization, and robustness. In
order to speed up the computations, we propose to use a multilevel
discretization technique. Our preliminary results suggest that the multilevel
discretization method improves solutions times by an order of magnitude
compared to existing methods without sacrificing stability or robustness.
