Remainders of generalised Taylor expansions and a priori bounds for rough differential equations

In this article we establish global a priori estimates on the solution of a generic rough differential equation driven by an $α$-Hölder path covering the full range of regularity $α\in(0,1)$, under the hypothesis of Lipschitz continuity (but neither boundedness nor coercivity) of specific combinations of the coefficients and their derivatives, called elementary differentials. Our main tool is a closed formula for the remainder of a generalised Taylor expansion in terms of products of gradients of the elementary differentials, which allows to take advantage of cancellations in an optimal way.