Controlled differential equations as rough integrals

TL;DR

This paper investigates controlled differential equations driven by rough paths with H"older continuity between 1/3 and 1/2, establishing existence, uniqueness, and stability of solutions interpreted via Gubinelli's rough integral framework.

Contribution

It extends the theory of rough differential equations to include unbounded drift terms with driving signals of lower regularity, using Gubinelli's controlled rough path approach.

Findings

Proves existence and uniqueness of solutions in the Gubinelli sense.

Establishes continuity of solutions with respect to initial conditions.

Provides estimates for the solution norms.

Abstract

We study controlled differential equations with unbounded drift terms, where the driving paths is $\nu$ - H\"older continuous for $\nu \in (\frac{1}{3},\frac{1}{2})$, so that the rough integral are interpreted in the Gubinelli sense \cite{gubinelli} for controlled rough paths. Similar to the rough differential equations in the sense of Lyons \cite{lyons98} or of Friz-Victoir \cite{friz}, we prove the existence and uniqueness theorem for the solution in the sense of Gubinelli, the continuity on the initial value, and the solution norm estimates.