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∈N of solutions to RDEs with unbounded drifts (ψn)n∈N. The penalisation ψn 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.