The accessibility problem for geometric rough differential equations

TL;DR

This paper demonstrates that solutions to geometric rough differential equations can be approximated by piecewise linear paths using geometric and rough calculus techniques, bridging classical and modern approaches.

Contribution

It introduces a geometric approach to approximate solutions of rough differential equations driven by geometric rough paths, combining Sussmann's orbit results with rough calculus on manifolds.

Findings

Terminal solutions can be obtained via piecewise linear driving paths.

The approach bridges classical orbit theory with modern rough calculus.

Provides a new method for approximating solutions to geometric rough differential equations.

Abstract

We show how to use geometric arguments to prove that the terminal solution to a rough differential equation driven by a geometric rough path can be obtained by driving the same equation by a piecewise linear path. For this purpose, we combine some results of the seminal work of Sussmann on orbits of vector fields with the rough calculus on manifolds developed by Cass, Litterer and Lyons.