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Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models

Author

Listed:
  • Christa Cuchiero

    (University of Vienna)

  • Francesca Primavera

    (University of Vienna)

  • Sara Svaluto-Ferro

    (University of Verona)

Abstract

We prove a universal approximation theorem that allows approximating continuous functionals of càdlàg (rough) paths uniformly in time and on compact sets of paths via linear functionals of their time-extended signature. Our main motivation to treat this question comes from signature-based models for finance that allow the inclusion of jumps. Indeed, as an important application, we define a new class of universal signature models based on an augmented Lévy process, which we call Lévy-type signature models. They extend continuous signature models for asset prices as proposed e.g. by Perez Arribas et al. (Proceedings of the First ACM International Conference on AI in Finance, ICAIF’20, Association for Computing Machinery, New York, 1–8, 2021) in several directions, while still preserving universality and tractability properties. To analyse this, we first show that the signature process of a generic multivariate Lévy process is a polynomial process on the extended tensor algebra and then use this for pricing and hedging approaches within Lévy-type signature models.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:finsto:v:29:y:2025:i:2:d:10.1007_s00780-025-00557-5
    DOI: 10.1007/s00780-025-00557-5
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    References listed on IDEAS

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    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    2. Erdinc Akyildirim & Matteo Gambara & Josef Teichmann & Syang Zhou, 2022. "Applications of Signature Methods to Market Anomaly Detection," Papers 2201.02441, arXiv.org, revised Feb 2022.
    3. Terry Lyons & Sina Nejad & Imanol Perez Arribas, 2020. "Non-parametric Pricing and Hedging of Exotic Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(6), pages 457-494, November.
    4. Hao Zhou, 2003. "Itô Conditional Moment Generator and the Estimation of Short-Rate Processes," Journal of Financial Econometrics, Oxford University Press, vol. 1(2), pages 250-271.
    5. Huyên Pham, 2000. "On quadratic hedging in continuous time," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(2), pages 315-339, April.
    6. Schweizer, Martin, 1991. "Option hedging for semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 339-363, April.
    7. Christa Cuchiero & Guido Gazzani & Janka Möller & Sara Svaluto‐Ferro, 2025. "Joint calibration to SPX and VIX options with signature‐based models," Mathematical Finance, Wiley Blackwell, vol. 35(1), pages 161-213, January.
    8. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.
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    More about this item

    Keywords

    Càdlàg rough paths; Signature; Universal approximation theorems; Financial modelling with jumps;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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