Penalisation techniques for one-dimensional reflected rough differential equations

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 continuous with respect to the Lebesgue measure.