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

Ariel Neufeld, Philipp Schmocker, Sizhou Wu

TL;DR

The paper develops a randomized deep splitting framework that uses random neural networks to solve high-dimensional nonlinear PDEs and PIDEs with jumps, including infinite-activity cases. It provides a full error analysis proving convergence to the unique viscosity solution and derives explicit bounds for both deterministic and random-network variants, with the random version offering controlled generalization error. Numerical experiments in pricing derivatives under default risk demonstrate the method scales to dimensions up to $d=10^4$, delivering solutions in seconds and outperforming deterministic DS and MLP approaches, especially for PIDEs with nonlocal terms. The results establish a scalable, provably convergent approach for high-dimensional nonlinear PIDE problems, combining rigorous theory with practical efficiency for financial engineering applications.

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.

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

TL;DR

The paper develops a randomized deep splitting framework that uses random neural networks to solve high-dimensional nonlinear PDEs and PIDEs with jumps, including infinite-activity cases. It provides a full error analysis proving convergence to the unique viscosity solution and derives explicit bounds for both deterministic and random-network variants, with the random version offering controlled generalization error. Numerical experiments in pricing derivatives under default risk demonstrate the method scales to dimensions up to , delivering solutions in seconds and outperforming deterministic DS and MLP approaches, especially for PIDEs with nonlocal terms. The results establish a scalable, provably convergent approach for high-dimensional nonlinear PIDE problems, combining rigorous theory with practical efficiency for financial engineering applications.

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.
Paper Structure (25 sections, 15 theorems, 171 equations, 4 tables, 1 algorithm)

This paper contains 25 sections, 15 theorems, 171 equations, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Derivation of the algorithm
  5. Temporal discretization
  6. Feynman-Kac representation
  7. Time discretization of SDEs with jumps
  8. Deep Splitting Algorithm with Deterministic Neural Networks
  9. Deep Splitting Algorithm with Random Neural Networks
  10. Pseudocode
  11. Setting and main results
  12. Assumptions and Setting
  13. Main Results
  14. Numerical examples
  15. Pricing under default risk without jumps
  16. ...and 10 more sections

Key Result

Theorem 3.9

For $\gamma\in(0,\infty)$, let $\rho \in \overline{C_b(\mathbb{R})}^\gamma$ be weakly differentiable and non-polynomial. Moreover, let Assumptions assumption Lip and growth--assumption time Holder hold, and let Assumption AssEtaMoments hold with some $c \in \mathbb{N}_0 \cap [\gamma,\infty)$. Then, As a consquence, we therefore conclude for every $d,N,M,\mathcal{M}\in \mathbb{N}$ and $\delta,\var

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 3.7
  • Theorem 3.9
  • Remark 3.10
  • Theorem 3.12
  • Corollary 3.13
  • Lemma 5.1
  • proof
  • Corollary 5.2
  • proof
  • ...and 23 more