Rough differential equations and reduced rough paths

TL;DR

The paper develops a streamlined framework for rough differential equations driven by reduced rough paths in the roughness range $\frac{1}{3}<\alpha\le\frac{1}{2}$. It defines reduced rough paths and reduced controlled rough paths, proves a key upper bound for compositions with regular functions, and constructs the reduced rough integral via the sewing lemma. Using a Banach fixed-point argument in the space of reduced X-controlled rough paths, the authors establish existence and uniqueness of solutions, with local results extended globally under stronger regularity. This provides a self-contained alternative to classical rough path methods, with potential advantages for Gaussian processes and fractional Brownian motion scenarios where the antisymmetric second-level information is unnecessary or problematic.

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 $\frac{1}{3} < α\leq \frac{1}{2}$. 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.

Theorems & Definitions (22)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Proposition 2.6
• proof
• Corollary 2.7
• proof