  In this paper, we introduce various machine learning solvers for (coupled)
forward-backward systems of stochastic differential equations (FBSDEs) driven
by a Brownian motion and a Poisson random measure. We provide a rigorous
comparison of the different algorithms and demonstrate their effectiveness in
various applications, such as cases derived from pricing with jumps and
mean-field games. In particular, we show the efficiency of the deep-learning
algorithms to solve a coupled multi-dimensional FBSDE system driven by a
time-inhomogeneous jump process with stochastic intensity, which describes the
Nash equilibria for a specific mean-field game (MFG) problem for which we also
provide the complete theoretical resolution. More precisely, we develop an
extension of the MFG model for smart grids introduced in Alasseur, Campi,
Dumitrescu and Zeng (Annals of Operations Research, 2023) to the case when the
random jump times correspond to the jump times of a doubly Poisson process. We
first provide an existence result of an equilibria and derive its semi-explicit
characterization in terms of a system of FBSDEs in the linear-quadratic
setting. We then compare the MFG solution to the optimal strategy of a central
planner and provide several numerical illustrations using the deep-learning
solvers presented in the first part of the paper.
