In this paper, we present a randomized extension of the deep splitting algorithm introduced in [Beck, Becker, Cheridito, Jentzen, and Neufeld (2021)] using random neural networks suitable to approximately solve both high-dimensional nonlinear parabolic PDEs and PIDEs with jumps having (possibly) infinite activity. We provide a full error analysis of our so-called random deep splitting method. In particular, we prove that our random deep splitting method converges to the (unique viscosity) solution of the nonlinear PDE or PIDE under consideration. Moreover, we empirically analyze our random deep splitting method by considering several numerical examples including both nonlinear PDEs and nonlinear PIDEs relevant in the context of pricing of financial derivatives under default risk. In particular, we empirically demonstrate in all examples that our random deep splitting method can approximately solve nonlinear PDEs and PIDEs in 10'000 dimensions within seconds.

翻译：本文提出了一种基于随机神经网络的深度分裂算法的随机扩展，该算法适用于近似求解具有（可能）无限活跃跳跃的高维非线性抛物型偏微分方程和偏积分微分方程。我们对所提出的随机深度分裂方法进行了完整的误差分析。特别地，我们证明了该随机深度分裂方法收敛于所考虑非线性偏微分方程或偏积分微分方程的（唯一黏性）解。此外，我们通过多个数值算例对随机深度分裂方法进行了实证分析，包括在违约风险下金融衍生品定价背景下相关的非线性偏微分方程和非线性偏积分微分方程。实证结果表明，在所有算例中，我们的随机深度分裂方法均能在数秒内近似求解万维空间中的非线性偏微分方程和偏积分微分方程。