We propose a deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and high-dimensional forward-backward stochastic differential equations with jumps (FBSDEJs), where the jump-diffusion process are derived by a Brownian motion and an independent compensated Poisson random measure. In this novel algorithm, a pair of deep neural networks for the approximations of the gradient and the integral kernel is introduced in a crucial way based on deep FBSDE method. To derive the error estimates for this deep learning algorithm, the convergence of Markovian iteration, the error bound of Euler time discretization, and the simulation error of deep learning algorithm are investigated. Two numerical examples are provided to show the efficiency of this proposed algorithm.

翻译：我们提出了一种深度学习算法，用于求解高维抛物型积分-微分方程（PIDE）以及带跳的高维正倒向随机微分方程（FBSDEJ），其中跳扩散过程由布朗运动与独立的补偿泊松随机测度导出。在该创新算法中，基于深度FBSDE方法，我们关键性地引入了一对深度神经网络，分别用于近似梯度项与积分核函数。为推导该深度学习算法的误差估计，本文研究了马尔可夫迭代的收敛性、欧拉时间离散化的误差界以及深度学习算法的仿真误差。最后通过两个数值算例验证了所提算法的有效性。