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.

该论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，用于近似解决具有（可能）无线活动的高维非线性抛物线 PDE 和跳跃 PIDE。我们对所谓的随机深度分裂方法进行了全面的误差分析，并证明我们的方法在考虑的非线性 PDE 或 PIDE 的唯一黏度解上收敛。此外，我们通过考虑几个在违约风险下定价金融衍生品方面的相关非线性 PDE 和非线性 PIDE 的数值例子来经验性地分析我们的随机深度分裂方法。特别地，我们在所有例子中经验性地证明我们的随机深度分裂方法可以在秒级内近似解决 10,000 维的非线性 PDE 和 PIDE。