Volterra equations driven by rough signals 3: Probabilistic construction of the Volterra rough path for fractional Brownian motions (2202.05076v1)

Abstract: 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 $|t-s|^{-\gamma}$ for some $\gamma\geq 0$. The construction is done in both the Stratonovich and It^o 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.