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Global universality via discrete-time signatures

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  • Mihriban Ceylan
  • David J. Promel

Abstract

We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition on the underlying weight function. As an application, we show that piecewise linear interpolations of Brownian motion satisfies this integrability condition. Consequently, we obtain $L^p$-approximation results for path-dependent functionals, random ordinary differential equations, and stochastic differential equations driven by Brownian motion.

Suggested Citation

  • Mihriban Ceylan & David J. Promel, 2026. "Global universality via discrete-time signatures," Papers 2603.09773, arXiv.org.
    • Handle: RePEc:arx:papers:2603.09773
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    References listed on IDEAS

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    1. Zdzisław Denkowski & Stanisław Migórski & Nikolas S. Papageorgiou, 2003. "An Introduction to Nonlinear Analysis: Theory," Springer Books, Springer, number 978-1-4419-9158-4, January.
    2. Christian Bayer & Luca Pelizzari & John Schoenmakers, 2025. "Primal and dual optimal stopping with signatures," Finance and Stochastics, Springer, vol. 29(4), pages 981-1014, October.
    3. Christian Bayer & Luca Pelizzari & John Schoenmakers, 2023. "Primal and dual optimal stopping with signatures," Papers 2312.03444, arXiv.org, revised Feb 2025.
    4. Fermanian, Adeline, 2021. "Embedding and learning with signatures," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    5. Terry Lyons & Sina Nejad & Imanol Perez Arribas, 2019. "Numerical Method for Model-free Pricing of Exotic Derivatives in Discrete Time Using Rough Path Signatures," Applied Mathematical Finance, Taylor & Francis Journals, vol. 26(6), pages 583-597, November.
    6. Mihriban Ceylan & David J. Promel, 2025. "Global universal approximation with Brownian signatures," Papers 2512.16396, arXiv.org.
    7. Christa Cuchiero & Francesca Primavera & Sara Svaluto-Ferro, 2025. "Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models," Finance and Stochastics, Springer, vol. 29(2), pages 289-342, April.
    8. Philipp Doersek & Josef Teichmann, 2010. "A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations," Papers 1011.2651, arXiv.org.
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