Volterra equations driven by rough signals 2: higher order expansions

TL;DR

This paper extends rough path theory for Volterra equations to handle more irregular noise and kernels, using algebraic structures like rooted trees to solve these complex equations.

Contribution

It introduces a new algebraic framework for Volterra rough paths, enabling analysis of more irregular signals and kernels in Volterra equations.

Findings

Extended rough path theory to more irregular signals.

Developed a rooted tree algebraic description.

Solved rough Volterra equations with increased irregularity.

Abstract

We extend the recently developed rough path theory for Volterra equations from (Harang and Tindel, 2019) to the case of more rough noise and/or more singular Volterra kernels. It was already observed in (Harang and Tindel, 2019) that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of "non-geometric rough paths" developed in (Gubinelli, 2010) and (Hairer and Kelly, 2015) we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough volterra equations in driven by more irregular signals.