Paper: arXiv 2503.16696

Authors: Anna P. Kwossek, David J. Prömel, Josef Teichmann

Abstract

We identify various classes of neural networks that are able to approximate continuous functions locally uniformly subject to fixed global linear growth constraints. For such neural networks the associated neural stochastic differential equations can approximate general stochastic differential equations, both of Itô diffusion type, arbitrarily well. Moreover, quantitative error estimates are derived for stochastic differential equations with sufficiently regular coefficients.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical proof establishing the universal approximation property for Neural SDEs under explicit global linear growth constraints using weighted function spaces. While mathematically strong and foundational for quantitative finance modeling, it is entirely theoretical with no empirical simulations or code provided.

Research Flowchart

  flowchart TD
    A[Research Goal: Prove Universal Approximation for Neural SDEs] --> B[Inputs: Target Itô Diffusions & NN Architectures]
    B --> C[Methodology: Apply Global Linear Growth Constraints to NNs]
    C --> D[Approximation: Fit Continuous Functions Locally Uniformly]
    D --> E[Error Analysis: Derive Quantitative Error Bounds for Regular Coefficients]
    E --> F[Outcome: Neural SDEs Universally Approximate General SDEs]

Cited References (26)

  1. Global universal approximation of functional input maps on weighted spaces — → 2306.03303
  2. Non-adversarial training of Neural SDEs with signature kernel scores — → 2305.16274
  3. FuNVol: A Multi-Asset Implied Volatility Market Simulator using Functional Principal Components and Neural SDEs — → 2303.00859
  4. Hedging Option Books Using Neural-SDE Market Models (2022) — link
  5. Efficient and Accurate Gradients for Neural SDEs (2021) — link
  6. Arbitrage-Free Neural-SDE Market Models (2021) — link
  7. A Generative Adversarial Network Approach to Calibration of Local Stochastic Volatility Models (2020) — link
  8. Scalable Gradients for Stochastic Differential Equations (2020) — link
and 18 more
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  2. Neural Jump Stochastic Differential Equations (2019) — link
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  9. Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit (2019)
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  18. Machine learning methods for pricing financial derivatives

3 of 26 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.