Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs

TL;DR

This paper introduces a randomized deep splitting algorithm for high-dimensional nonlinear PDEs and PIDEs, providing full error analysis and demonstrating rapid, accurate solutions in up to 10,000 dimensions.

Contribution

It presents a novel randomized extension of the deep splitting method with comprehensive error analysis and empirical validation for complex high-dimensional problems.

Findings

Converges to the unique viscosity solution of PDEs and PIDEs.

Solves high-dimensional problems in seconds.

Effective for nonlinear PDEs and PIDEs in finance.

Abstract

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.