We propose a new deep learning algorithm for solving high-dimensional parabolic integro- differential equations (PIDEs) and forward-backward stochastic differential equations with jumps (FBSDEJs). This novel algorithm can be viewed as an extension and generalization of the DBDP2 scheme and a dynamic programming version of the forward-backward algorithm proposed recently for high-dimensional semilinear PDEs and semilinear PIDEs, respectively. Different from the DBDP2 scheme for semilinear PDEs, our algorithm approximate simultaneously the solution and the integral kernel by deep neural networks, while the gradient of the solution is approximated by numerical differential techniques. The related error estimates for the integral kernel approximation play key roles in deriving error estimates for the novel algorithm. Numerical experiments confirm our theoretical results and verify the effectiveness of the proposed methods.

翻译：本文提出了一种用于求解高维抛物型积分-微分方程（PIDEs）及带跳跃的正倒向随机微分方程（FBSDEJs）的新型深度学习算法。该算法可视为DBDP2方案的扩展与推广，同时也是近期分别针对高维半线性偏微分方程和半线性PIDEs所提出的前向-后向算法的动态规划版本。与针对半线性偏微分方程的DBDP2方案不同，本算法通过深度神经网络同时逼近解函数与积分核，而解的梯度则通过数值微分技术进行近似。积分核逼近的相关误差估计在推导新算法的误差估计中起着关键作用。数值实验验证了理论结果，并证实了所提方法的有效性。