  In this paper, we propose a numerical methodology for finding the closed-loop
Nash equilibrium of stochastic delay differential games through deep learning.
These games are prevalent in finance and economics where multi-agent
interaction and delayed effects are often desired features in a model, but are
introduced at the expense of increased dimensionality of the problem. This
increased dimensionality is especially significant as that arising from the
number of players is coupled with the potential infinite dimensionality caused
by the delay. Our approach involves parameterizing the controls of each player
using distinct recurrent neural networks. These recurrent neural network-based
controls are then trained using a modified version of Brown's fictitious play,
incorporating deep learning techniques. To evaluate the effectiveness of our
methodology, we test it on finance-related problems with known solutions.
Furthermore, we also develop new problems and derive their analytical Nash
equilibrium solutions, which serve as additional benchmarks for assessing the
performance of our proposed deep learning approach.
