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Volterra Equations Driven by Rough Signals 3: Probabilistic Construction of the Volterra Rough Path for Fractional Brownian Motions

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  • Fabian Harang

    (BI Norwegian Business School)

  • Samy Tindel

    (Purdue University)

  • Xiaohua Wang

    (Purdue University)

Abstract

Based on the recent development of the framework of Volterra rough paths (Harang and Tindel in Stoch Process Appl 142:34–78, 2021), we consider here the probabilistic construction of the Volterra rough path associated to the fractional Brownian motion with $$H>\frac{1}{2}$$ H > 1 2 and for the standard Brownian motion. The Volterra kernel k(t, s) is allowed to be singular, and behaving similar to $$|t-s|^{-\gamma }$$ | t - s | - γ for some $$\gamma \ge 0$$ γ ≥ 0 . The construction is done in both the Stratonovich and Itô senses. It is based on a modified Garsia–Rodemich–Romsey lemma which is of interest in its own right, as well as tools from Malliavin calculus. A discussion of challenges and potential extensions is provided.

Suggested Citation

  • Fabian Harang & Samy Tindel & Xiaohua Wang, 2024. "Volterra Equations Driven by Rough Signals 3: Probabilistic Construction of the Volterra Rough Path for Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 37(1), pages 307-351, March.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-023-01251-y
    DOI: 10.1007/s10959-023-01251-y
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    References listed on IDEAS

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    1. Hu, Yaozhong & Le, Khoa, 2013. "A multiparameter Garsia–Rodemich–Rumsey inequality and some applications," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3359-3377.
    2. Christian Bayer & Peter K. Friz & Paul Gassiat & Jorg Martin & Benjamin Stemper, 2020. "A regularity structure for rough volatility," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 782-832, July.
    3. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    4. Harang, Fabian A. & Tindel, Samy, 2021. "Volterra equations driven by rough signals," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 34-78.
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