Parisian Excursion With Capital Injection Under Draw-down Reflected Lévy Insurance Risk Process

The Parisian ruin problem for spectrally negative Levy risk processes have been extensively studied in recent years.In this paper,the spectrally negative Levy processes and its scale functions are studied.The structure is as follows:The literature review of this paper is given in the first Chapter.Firstly,the preparatory knowledge of this paper is given In Chapter 2.We define the spectrally negative Levy processes and its scale functions.Then,motivated by Wang and Zhou(2019),we define the draw-down time,the reflected process of the spectrally negative Levy processes and the draw-down reflected process of the spectrally negative Levy processes.We also define the first up-crossing and down-crossing time and Parisian ruin time of the draw-down reflected process of the spectrally negative Levy processes.Then,we define the total amount of capital injections made until time t for the draw-down reflected process.In Chapter 3,we study the Parisian ruin problem for the draw-down reflected process of the spectrally negative Levy risk processes.Some classical results on the spectrally negative Levy process reflected from the infimum are generalized to the process with draw-down reflection.Using an excursion-theoretical approach,for such a reflected process we solve a two-sides exit problem.We find the Laplace transform of the twosides exiting time.Then,we evaluate the joint Laplace transform expression of the twosides exit related to general drawdown time under the spectral negative Levy reflection process.These conclusions are expressed by the scale function of Levy 's risk process.