The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs.

中文翻译：

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耦合FBSDE的深BSDE方法的收敛性

最近提出的数值算法，深BSDE方法，在求解高维前后倒向随机微分方程（FBSDE）和抛物线偏微分方程（PDE）方面显示了卓越的性能。本文为耦合FBSDE的一般情况下的深度BSDE方法奠定了理论基础。特别地，提供了解决方案的后验误差估计，并证明了在神经网络具有通用逼近能力的情况下，该误差收敛至零。数值结果表明了所分析算法在求解高维耦合FBSDE中的准确性。