  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.
