Invariance for Rough Differential Equations

TL;DR

This paper extends classical invariance conditions from stochastic differential equations driven by Brownian motion to rough differential equations, including those driven by fractional Brownian motion, providing new theoretical insights and comparison results.

Contribution

It generalizes invariance criteria to rough differential equations and offers a comparison theorem, broadening the applicability of invariance analysis in stochastic calculus.

Findings

Extended invariance conditions to rough differential equations.

Provided a comparison theorem for rough differential equations.

Analyzed the special case of fractional Brownian motion.

Abstract

In 1990, in It\^o's stochastic calculus framework, Aubin and Da Prato established a necessary and sufficient condition of invariance of a nonempty compact or convex subset $C$ of $\mathbb R^d$ ($d\in\mathbb N^*$) for stochastic differential equations (SDE) driven by a Brownian motion. In Lyons rough paths framework, this paper deals with an extension of Aubin and Da Prato's results to rough differential equations. A comparison theorem is provided, and the special case of differential equations driven by a fractional Brownian motion is detailed.