# Penalisation techniques for one-dimensional reflected rough differential equations

## TL;DR

This paper introduces a penalisation method to solve one-dimensional reflected rough differential equations (RDEs) with irregular boundaries, establishing existence, convergence rates, and absolute continuity of solutions' laws.

## Contribution

It develops a novel penalisation approach for reflected RDEs, providing existence, convergence speed, and law regularity results, including a Doss-Sussmann representation.

## Key findings

- Established existence of solutions via penalisation
- Derived convergence speed of penalised solutions
- Proved absolute continuity of the law of solutions at positive times

## Abstract

In this paper we solve real-valued rough differential equations (RDEs) reflected on an irregular boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{n\in\mathbb{N}}$ of solutions to RDEs with unbounded drifts $(\psi_n)_{n\in\mathbb{N}}$. The penalisation $\psi_n$ increases with $n$. Along the way, we thus also provide an existence theorem and a Doss-Sussmann representation for RDEs with a drift growing at most linearly. In addition, a speed of convergence of the sequence of penalised paths to the reflected solution is obtained. \\ We finally use the penalisation method to prove that the law at time $t>0$ of some reflected Gaussian RDE is absolutely contiuous with respect to the Lebesgue measure.