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

Fabian Harang (), Samy Tindel () and Xiaohua Wang ()
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Fabian Harang: BI Norwegian Business School
Samy Tindel: Purdue University
Xiaohua Wang: Purdue University

Journal of Theoretical Probability, 2024, vol. 37, issue 1, 307-351

Abstract: 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.

Keywords: Volterra equation; Signature; Rough path; Malliavin calculus; Fractional Brownian motion; Primary: 60L20; 60L10; Secondary: 60H07; 60H05; 60G22 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01251-y

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