Penalisation techniques for one-dimensional reflected rough differential equations
- Resource Type
- Working Paper
- Authors
- Richard, Alexandre; Tanré, Etienne; Torres, Soledad
- Source
- Subject
- Mathematics - Probability
34F05, 60G15, 60H10
- Language
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.