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:我们建议了一种深层次的学习算法,用于解决高维抛射分化方程式(PIDES)和高维的前向后向前向前向偏移方程式(FBSDEJs ), 跳跃扩散过程由布朗运动和独立补偿的Poisson随机测量法产生。在这个小说算法中,以深深深FBSDE方法为基础,以关键的方式引入了两组深层的梯度近似神经网络和整体内核。要得出这一深深层次学习方程式的误差估计值、Markovian迭代的趋同、Euler时间分解的误差以及深层次学习方程式的模拟误差。提供了两个数字例子,以显示这一拟议算法的效率。