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

Abstract

Based on the recent development of the framework of Volterra rough paths [15], we consider here the probabilistic construction of the Volterra rough path associated to the fractional Brownian motion with 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| −γ for some γ ≥ 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.