Penalisation techniques for one-dimensional reflected rough differential equations

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.