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

TL;DR

This paper investigates the sensitivity of solutions to rough differential equations by analyzing the regularity of the Itô map, demonstrating that it is Hölder or Lipschitz continuous under weaker conditions than previously known.

Contribution

The paper introduces a novel approach using the Omega lemma to establish the Hölder and Lipschitz continuity of the Itô map with relaxed regularity assumptions.

Findings

Itô map is Hölder continuous with respect to all parameters.

Itô map is Lipschitz continuous under certain conditions.

Results unify and weaken previous regularity hypotheses.

Abstract

The It{\^o} map assigns the solution of a Rough Differential Equation, a generalization of an Ordinary Differential Equation driven by an irregular path, when existence and uniqueness hold. By studying how a path is transformed through the vector field which is integrated, we prove that the It{\^o} map is H{\"o}lder or Lipschitz continuous with respect to all its parameters. This result unifies and weaken the hypotheses of the regularity results already established in the literature.