Paper: arXiv 2509.26535

Authors: Yun Zhao, Harry Zheng

Abstract

We propose an approach to applying neural networks on linear parabolic variational inequalities. We use loss functions that directly incorporate the variational inequality on the whole domain to bypass the need to determine the stopping region in advance and prove the existence of neural networks whose losses converge to zero. We also prove the functional convergence in the Sobolev space. We then apply our approach to solving an optimal investment and stopping problem in finance. By leveraging duality, we convert the nonlinear HJB-type variational inequality of the primal problem into a linear variational inequality of the dual problem and prove the convergence of the primal value function from the dual neural network solution, an outcome made possible by our Sobolev norm analysis. We illustrate the versatility and accuracy of our method with numerical examples for both power and non-HARA utilities as well as high-dimensional American put option pricing. Our results underscore the potential of neural networks for solving variational inequalities in optimal stopping and control problems.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous mathematical proofs of deep neural network functional convergence in Sobolev space for variational inequalities using trace theorems and duality. While it presents strong theoretical contributions and tests on pricing models, it lacks extensive empirical backtesting on real market data, placing it in the theoretical domain.

Research Flowchart

  flowchart TD
    A[Goal: Solve Parabolic Variational Inequalities via Neural Networks] --> B[Formulate Loss Function incorporating VI over whole domain]
    C[Problem Inputs: Utility Functions & American Option Parameters] --> D[Convert Primal HJB VI to Dual Linear VI]
    B --> E[Train Dual Neural Network Model]
    D --> E
    E --> F[Prove Theoretical Loss & Sobolev Space Convergence]
    F --> G[Outcomes: Value Function Convergence & High-Dim Option Pricing]

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0 of 33 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.