Sizhou Wu

Ariel Neufeld, Philipp Schmocker, Sizhou Wu

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.

Yan Ding, Sizhou Wu, Ying Zhang

We study the temporal-spatial regularity properties of tamed Euler approximations for Lévy-driven SDEs with superlinearly growing drift and diffusion coefficients. We first introduce a novel tamed Euler-type scheme and establish its strong convergence. We then derive temporal-spatial regularity estimates with respect to the initial value, the initial time, and the evaluation time. In particular, we obtain a stability estimate for fixed step size and a corresponding continuity estimate in the vanishing step-size regime. Numerical experiments are presented to support the theoretical results.