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Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations

  • Wonjae Lee
  • Jiuk Jang
  • Hyungbin Park
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Abstract

This paper studies Wong--Zakai convergence and Fourier neural operator learning for stochastic Volterra integral equations (SVIEs) describing random systems with memory. A Wong--Zakai theorem establishes convergence of pathwise solutions driven by piecewise linear approximations of Brownian motion to the corresponding SVIE solution. This convergence provides a foundation for using mirror-padded Fourier neural operators to learn the solution operator mapping initial conditions and driving noise to entire solution trajectories. Numerical experiments demonstrate competitive prediction accuracy and generalization to temporal grids finer than those used for training, while assessing the influence of padding and computational cost. As a follow-up to \citet{LKP:26}, this study extends their operator-learning framework to SVIEs through separate convergence and approximation analyses tailored to convolutional memory.

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Cite

BibTeXRIS
BibTeX
@misc{lee2026wongzakai,
  title = {{Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations}},
  author = {Wonjae Lee and Jiuk Jang and Hyungbin Park},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18939},
  eprint = {2610.18939},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18939},
}
RIS
TY  - UNPB
TI  - Wong--Zakai Convergence and FNO Learning for Stochastic Volterra Equations
AU  - Wonjae Lee
AU  - Jiuk Jang
AU  - Hyungbin Park
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18939
UR  - https://arxiv.org/abs/2610.18939
AB  - This paper studies Wong--Zakai convergence and Fourier neural operator learning for stochastic Volterra integral equations (SVIEs) describing random systems with memory. A Wong--Zakai theorem establishes convergence of pathwise solutions driven by piecewise linear approximations of Brownian motion to the corresponding SVIE solution. This convergence provides a foundation for using mirror-padded Fourier neural operators to learn the solution operator mapping initial conditions and driving noise to entire solution trajectories. Numerical experiments demonstrate competitive prediction accuracy and generalization to temporal grids finer than those used for training, while assessing the influence of padding and computational cost. As a follow-up to \citet{LKP:26}, this study extends their operator-learning framework to SVIEs through separate convergence and approximation analyses tailored to convolutional memory.
ER  -