# On the definition of a solution to a rough differential equation

## TL;DR

This paper proves that two different definitions of solutions to rough differential equations are equivalent, offering a new perspective on the algebraic structure underlying rough path theory.

## Contribution

It provides an elementary proof of the equivalence between Davie's and Bailleul's solution concepts, connecting different approaches in rough path theory.

## Key findings

- Davie's and Bailleul's definitions of solutions are equivalent
- The proof offers a new elementary perspective
- Links to algebraic structures in rough path theory

## Abstract

We give an elementary proof that Davie's definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the deep algebraic insights of Cass and Weidner in their work (Tree algebras over topological vector spaces in rough path theory).