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Volterra equations driven by rough signals 3: Probabilistic construction of the Volterra rough path for fractional Brownian motions

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