Local discontinuous Galerkin method for nonlinear BSPDEs of Neumann boundary conditions with deep backward dynamic programming time-marching

Yixiang Dai, Yunzhang Li, Jing Zhang
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Abstract

This paper aims to present a local discontinuous Galerkin (LDG) method for solving backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of the proposed numerical scheme. Two numerical examples are provided to demonstrate the performance of the LDG method, where we incorporate a deep learning algorithm to address the challenge of the curse of dimensionality in backward stochastic differential equations (BSDEs). The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.
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采用深度后向动态程序设计时间行进的 Neumann 边界条件非线性 BSPDE 的局部非连续 Galerkin 方法
本文旨在提出一种局部非连续伽勒金(LDG)方法,用于解决具有纽曼边界条件的后向随机偏微分方程(BSPDEs)。我们建立了所提数值方案的 $L^2$ 稳定性和最优误差估计。我们提供了两个数值示例来证明 LDG 方法的性能,其中我们结合了深度学习算法来解决后向随机微分方程(BSDEs)中的维度诅咒难题。结果表明了 LDG 方法在处理具有 Neumann 边界条件的 BSPDEs 时的有效性和准确性。
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