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The It{\^o} map assigns the solution of a Rough Differential Equation, ageneralization of an Ordinary Differential Equation driven by an irregularpath, when existence and uniqueness hold. By studying how a path is transformedthrough the vector field which is integrated, we prove that the It{\^o} map isH{\"o}lder or Lipschitz continuous with respect to all its parameters. Thisresult unifies and weaken the hypotheses of the regularity results alreadyestablished in the literature.

Sensitivity of rough differential equations: an approach through the Omega lemma