Paper: arXiv 2405.05192

Authors: Ariel Neufeld, Philipp Schmocker, Sizhou Wu

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.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 6.8/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous, full error analysis and convergence proof for a new randomized deep splitting method on high-dimensional PDEs/PIDEs. It combines advanced mathematical theoretical foundation with remarkable high-dimensional (up to 10,000 dimensions) numerical demonstrations applied to derivative pricing under default risk.

Research Flowchart

  flowchart TD
    A[Goal: Solve High-Dim Nonlinear PDEs/PIDEs<br>Up to 10,000 Dimensions] --> B[Method: Develop Random Deep Splitting Method<br>Combining Splitting Algorithm & Random NNs]
    
    C[Inputs: Financial Derivative Models,<br>Default Risk & Infinite Activity Jumps] --> D
    
    B --> D[Theoretical Error Analysis:<br>Prove Convergence to Unique Viscosity Solution]
    B --> E[Empirical Validation:<br>Numerical Testing in High Dimensions]
    
    D --> F[Outcome: Guaranteed Theoretical Convergence<br>& Fast Solution in Seconds]
    E --> F

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1 of 135 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.