Rough differential equations and reduced rough paths

TL;DR

This paper proves existence and uniqueness of solutions for rough differential equations driven by reduced rough paths with low regularity, using a simplified fixed point approach in a Banach space of controlled paths.

Contribution

It introduces a streamlined, self-contained method for solving rough differential equations driven by reduced rough paths in the challenging roughness regime.

Findings

Established existence and uniqueness in the roughness regime 1/3 < α ≤ 1/2

Developed a fixed point argument in a Banach space of controlled paths

Provided a concise alternative to classical rough path theory

Abstract

This paper establishes the existence and uniqueness of solutions for rough differential equations driven by reduced rough paths with low regularity, specifically in the roughness regime $\frac{1}{3} < \alpha \leq \frac{1}{2}$. While the well-posedness of rough differential equations driven by classical rough paths in this regime is known, the reduced structure presents unique analytical challenges that fall outside the scope of classical theories. By formulating the problem within a suitably constructed Banach space of controlled paths, we implement a fixed point argument based on the Banach contraction principle. This approach provides a direct and self-contained proof, offering a clear and concise alternative to the more intricate machinery of the classical theory of rough differential equations. Our work thus provides a streamlined framework for analyzing this important class of rough equations.