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Functional It\^o-formula and Taylor expansions for non-anticipative maps of c\`adl\`ag rough paths

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Listed:
  • Christa Cuchiero
  • Xin Guo
  • Francesca Primavera

Abstract

We derive a functional It\^o-formula for non-anticipative maps of rough paths, based on the approximation properties of the signature of c\`adl\`ag rough paths. This result is a functional extension of the It\^o-formula for c\`adl\`ag rough paths (by Friz and Zhang (2018)), which coincides with the change of variable formula formulated by Dupire (2009) whenever the functionals' representations, the notions of regularity, and the integration concepts can be matched. Unlike these previous works, we treat the vertical (jump) pertubation via the Marcus transformation, which allows for incorporating path functionals where the second order vertical derivatives do not commute, as is the case for typical signature functionals. As a byproduct, we show that sufficiently regular non-anticipative maps admit a functional Taylor expansion in terms of the path's signature, leading to an important generalization of the recent results by Dupire and Tissot-Daguette (2022).

Suggested Citation

  • Christa Cuchiero & Xin Guo & Francesca Primavera, 2025. "Functional It\^o-formula and Taylor expansions for non-anticipative maps of c\`adl\`ag rough paths," Papers 2504.06164, arXiv.org.
    • Handle: RePEc:arx:papers:2504.06164
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    References listed on IDEAS

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    1. Bruno Dupire & Valentin Tissot-Daguette, 2022. "Functional Expansions," Papers 2212.13628, arXiv.org, revised Mar 2023.
    2. Nicolas Perkowski & David J. Promel, 2013. "Pathwise stochastic integrals for model free finance," Papers 1311.6187, arXiv.org, revised Jun 2016.
    3. Keller, Christian & Zhang, Jianfeng, 2016. "Pathwise Itô calculus for rough paths and rough PDEs with path dependent coefficients," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 735-766.
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