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

TL;DR

This paper develops a probabilistic method to construct Volterra rough paths driven by fractional Brownian motions with H>1/2 and standard Brownian motion, handling singular kernels using advanced stochastic calculus tools.

Contribution

It introduces a new probabilistic construction of Volterra rough paths for fractional Brownian motion with singular kernels, extending the rough path framework.

Findings

Construction in both Stratonovich and Itô senses.

Uses a modified Garsia-Rodemich-Romsey lemma and Malliavin calculus.

Addresses singular kernels similar to |t-s|^{-eta}.

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.