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

Regularisation by fractional noise for one-dimensional differential equations with nonnegative distributional drift

“…Note also that the case a = 1 corresponds to reflection above 0. In the frameworks of Young integrals and rough paths theory, well-posedness is established even for multiplicative noises in case X is one-dimensional [10,32], while uniqueness might fail as soon as the dimension is greater than 2 (see [19]).…”