Reflected rough differential equations

TL;DR

This paper introduces reflected rough differential equations driven by rough paths, establishing existence of solutions and exploring their relation to reflected stochastic differential equations driven by Brownian motion.

Contribution

It defines reflected rough differential equations for non-smooth domains and proves the existence of solutions, extending rough path theory to reflected settings.

Findings

Existence of solutions for reflected rough differential equations.

Connection between reflected stochastic and rough differential equations.

Extension of rough path theory to non-smooth boundary domains.

Abstract

In this paper, we study reflected differential equations driven by continuous paths with finite $p$-variation ($1\le p<2$) and $p$-rough paths ($2\le p<3$) on domains in Euclidean spaces whose boundaries may not be smooth. We define reflected rough differential equations and prove the existence of a solution. Also we discuss the relation between the solution to reflected stochastic differential equation and reflected rough differential equation when the driving process is a Brownian motion.