Ramification of Volterra-type Rough Paths

TL;DR

This paper develops a novel framework for stochastic Volterra equations by extending Rough Path theory to include Volterra-type Rough Paths, enabling global existence and uniqueness results.

Contribution

It introduces Volterra-type Rough Paths and convolution products, extending Rough Path theory to handle stochastic Volterra equations with a new analytical approach.

Findings

Established global existence and uniqueness for Volterra equations

Defined Volterra-type Rough Paths and convolution products

Extended Rough Path theory to Volterra kernels

Abstract

We extend the new approach introduced in arXiv:1912.02064v2 [math.PR] and arXiv:2102.10119v1 [math.PR] for dealing with stochastic Volterra equations using the ideas of Rough Path theory and prove global existence and uniqueness results. The main idea of this approach is simple: Instead of the iterated integrals of a path comprising the data necessary to solve any equation driven by that path, now iterated integral convolutions with the Volterra kernel comprise said data. This leads to the corresponding abstract objects called Volterra-type Rough Paths, as well as the notion of the convolution product, an extension of the natural tensor product used in Rough Path Theory.