Abstract
We study linear rough dierential equations and we solve perturbed linear rough dierential equation with a representation using the Duhamel principle. These results provide us with the key technical point to study the regularity of the dierential of the Ito map in a subsequent article.
Highlights
Linear Rough Differential Equations (RDE) have been considered by several authors since they are an essential tool for studying the derivative of the Itô map and its flow properties (See the bibliography in [19])
Their rough paths extensions which are the core of the theory, are solutions to linear RDE taking their values in tensor spaces, for which more precise results could be given
A p-rough path x of order is a path taking its values in T (V) with (i) The order satisfies ≥ p, where a is the integer part of a. (ii) For any t ∈ [0, T ] π0(xt) = 1 and xt is invertible in T (V). (iii) With (4), x hom ≺ Cω1/p
Summary
Linear Rough Differential Equations (RDE) have been considered by several authors since they are an essential tool for studying the derivative of the Itô map and its flow properties (See the bibliography in [19]). We define the notion of p-rough resolvent, which is an extension of the notion of multiplicative functionals [42,43] taking their values in a Banach algebra Chen series, and their rough paths extensions which are the core of the theory, are solutions to linear RDE taking their values in tensor spaces, for which more precise results could be given. We obtain an exponential representation of the p-rough resolvents, at least for a small time This provides us with extensions of the Magnus [7] and Chen-. In a subsequent article [19], we use these properties to show that the Itô map is differentiable with a Lipschitz or Hölder continuous Fréchet derivative
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