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.

翻译：本文针对由布朗运动与泊松随机测度驱动的（耦合）随机微分方程正倒向系统，提出了多种机器学习求解器。我们对不同算法进行了严格比较，并展示了它们在各类应用中的有效性，例如源自带跳跃定价与平均场博弈的案例。特别地，我们证明了深度学习算法在求解由具有随机强度的非齐次跳跃过程驱动的耦合多维FBSDE系统时的效率，该系统描述了特定平均场博弈问题的纳什均衡，我们同时给出了该问题的完整理论解。更精确地说，我们将Alasseur、Campi、Dumitrescu和Zeng（Annals of Operations Research, 2023）提出的智能电网MFG模型扩展至随机跳跃时间对应双重泊松过程跳跃时间的情形。我们首先证明了均衡解的存在性，并在线性二次设定下推导出其半显式表征——即一个FBSDE系统。随后，我们将MFG解与中央计划者的最优策略进行比较，并利用本文第一部分提出的深度学习求解器提供了若干数值算例。