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 is an element of N) of solutions to RDEs with unbounded drifts (psi(n))(x is an element of N). The penalisation 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.