The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations

TL;DR

This paper reviews a recent approach to rough path integration using classical Young integration and extends Schwartz's argument to establish existence, uniqueness, and continuity of solutions for rough differential equations driven by paths in nilpotent Lie or Butcher groups.

Contribution

It introduces a novel method connecting rough path theory with classical Young integration and extends Schwartz's argument to a broader class of rough differential equations.

Findings

Established existence and uniqueness of solutions

Proved continuity of solutions with respect to the driving path

Applicable to paths in nilpotent Lie and Butcher groups

Abstract

We give an overview of the recent approach to the integration of rough paths that reduces the problem to classical Young integration. As an application, we extend an argument of Schwartz to rough differential equations, and prove the existence, uniqueness and continuity of the solution, which is applicable when the driving path takes values in nilpotent Lie group or Butcher group.