Lipschitz-stability of Controlled Rough Paths and Rough Differential Equations

Abstract: We provide an account for the existence and uniqueness of solutions to rough differential equations under the framework of controlled rough paths. The case when the driving path is $\beta$-H\"older continuous, for $\beta>1/3$, is widely available in the literature. In its extension to the case when $\beta\leqslant1/3,$ a main challenge and missing ingredient is to show that controlled roughs paths are closed under composition with Lipschitz transformations. Establishing such a property precisely, which has a strong algebraic nature, is a main purpose of the present article.