Rough Volterra equations 1: the algebraic integration setting

TL;DR

This paper develops a framework using algebraic integration to solve Volterra equations driven by irregular signals, extending rough path theory to handle singular coefficients and fractional Brownian motion.

Contribution

It introduces a novel algebraic integration approach for Volterra equations, enabling solutions with irregular signals and singular coefficients, including fractional Brownian motion.

Findings

Global solutions for H"older exponent > 1/2

Local existence and uniqueness for H"older exponent in (1/3,1/2]

Applicable to fractional Brownian motion with H > 1/3

Abstract

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\"older exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with H\"older exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3.