Constructing general rough differential equations through flow approximations

On rough differential equations

We prove that the Itô map, that is the map that gives the solution of a differential equation controlled by a rough path of finite p-variation with p ∈ [2,3) is locally Lipschitz continuous in all its arguments and we give some sufficient conditions for global existence for non-bounded vector fields.

Approximations from Anywhere and General Rough Sets

Not all approximations arise from information systems. The problem of fitting approximations, subjected to some rules (and related data), to information systems in a rough scheme of things is known as the inverse problem. The inverse problem is more general than the duality (or abstract representation) problems and was introduced by the present author in her earlier papers. From the practical p...

Characterising small solutions in delay differential equations through numerical approximations

Perturbed linear rough differential equations

We study linear rough differential equations and we solve perturbed linear rough differential equation using the Duhamel principle. These results provide us with the key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach alge...

Sensitivity of rough differential equations∗

In the context of rough paths theory, we study the regularity of the Itô map with respect to the starting point, the coefficients and perturbation of the driving rough paths. In particular, we show that the Itô map is differentiable with a Hölder or Lipschitz continuous derivatives. With respect to the current literature on the subject, our proof rely on perturbation of linear Rough Differentia...