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Global universal approximation with Brownian signatures

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  • Mihriban Ceylan
  • David J. Promel

Abstract

We establish $L^p$-type universal approximation theorems for general and non-anticipative functionals on suitable rough path spaces, showing that linear functionals acting on signatures of time-extended rough paths are dense with respect to an $L^p$-distance. To that end, we derive global universal approximation theorems for weighted rough path spaces. We demonstrate that these $L^p$-type universal approximation theorems apply in particular to Brownian motion. As a consequence, linear functionals on the signature of the time-extended Brownian motion can approximate any $p$-integrable stochastic process adapted to the Brownian filtration, including solutions to stochastic differential equations.

Suggested Citation

  • Mihriban Ceylan & David J. Promel, 2025. "Global universal approximation with Brownian signatures," Papers 2512.16396, arXiv.org.
    • Handle: RePEc:arx:papers:2512.16396
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    File URL: http://arxiv.org/pdf/2512.16396
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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, December.
    2. Christian Bayer & Luca Pelizzari & John Schoenmakers, 2023. "Primal and dual optimal stopping with signatures," Papers 2312.03444, arXiv.org, revised Feb 2025.
    3. 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.
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    5. 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.
    6. Erhan Bayraktar & Qi Feng & Zhaoyu Zhang, 2022. "Deep Signature Algorithm for Multi-dimensional Path-Dependent Options," Papers 2211.11691, arXiv.org, revised Jan 2024.
    7. Christa Cuchiero & Philipp Schmocker & Josef Teichmann, 2023. "Global universal approximation of functional input maps on weighted spaces," Papers 2306.03303, arXiv.org, revised Dec 2025.
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    Cited by:

    1. Mihriban Ceylan & David J. Promel, 2026. "Global universality via discrete-time signatures," Papers 2603.09773, arXiv.org.

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