Parisian ruin for a refracted L\'evy process

In this paper, we investigate Parisian ruin for a L\'evy surplus process with an adaptive premium rate, namely a refracted L\'evy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al. (2013) for the probability of Parisian ruin of a standard L\'evy insurance risk process. Despite the more general setup considered here, our main result is as compact and has a similar structure. Examples are provided.