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：本文提出一种基于随机神经网络的深度分裂算法随机扩展（原始算法见[Beck, Becker, Cheridito, Jentzen, and Neufeld (2021)]），适用于近似求解高维非线性抛物型偏微分方程及具有（可能）无限活性的跳跃型偏微分-积分方程。我们对该称为随机深度分裂方法进行了完整误差分析，特别证明了该方法收敛于所考虑非线性偏微分方程或偏微分-积分方程的（唯一黏性）解。此外，通过若干数值算例（包括违约风险金融衍生品定价背景下的非线性偏微分方程与非线性偏微分-积分方程）进行实证分析，在所有算例中均表明该方法可在秒级内近似求解10,000维非线性偏微分方程与偏微分-积分方程。