On The Refracted Lévy Risk Processes With Parisian Delays

In this paper,we investigate Parisian ruin for a Lévy process with an adaptive premium rate,namely a refracted Lévy risk process.In this paper,we generalize the risk processes from a special refracted Lévy process with b = 0 which is studied in Lkabous et al.(2017)to a general refracted Lévy process with b?0.Using the scale functions,the Parisian ruin probability is given.The paper is organized as follows.Chapter one is an introduction.In this part,we mainly introduce the concept of Parisian ruin,refracted Lévy process and Parisian ruin time.The second chapter is divided into three parts.In the first part,spectrally negative Lévy processes,scale functions and fluctuation identities are introduced.Refracted Lévy risk processes are given in the second part.In the last part,we give some lemmas.In Chapter 3,we present and prove the main theorem and lemmas.At the last chapter,four examples are given.