Study On Two Problems In Spectrally Negative Lévy Processes

The exit problem of spectrally negative Lévy processes and its application in insur-ance risk model have been extensively studied in recent years.In this paper,the spectrally negative Lévy processes and its scale functions are studied.The structure is as follows:The literature review and preparatory knowledge of this paper are given in the first Chapter.In Chapter 2,motivated by Albrecher et al.(2016)and Li et al.(2017),we define the first up-crossing and down-crossing tim e and.general draw-down time of spectrally negative Lévy processes,and also define some stopping times of spectrally negative Lévy processes concerned with the arrival times of an independent Poisson process with rate of ?.A series of Laplace transform identities for the exit problem of spectrally negative Lévy processes related to the above-mentioned times are obtained by using the relevant knowledge of the excursion theory.The main theorem of this chapter is given by theorem 2.7,which is composed of the exit identities expressed in terms of the scale functions.By setting the value of the general draw-down function,we can get the relevant corollary of the theorem in this chapter.In the special case that the general draw-down function equal to 0 in the domain of[0,?),we can know that the results in this chapter are consistent with theorem 3.1 of Albrecher et al.(2016).Chapter 3 revisits the spectrally negative Lévy risk process embedded with the gener-al tax structure introduced in Kyprianou and Zhou(2009).We determine the surplus level to start taxation which maximises the performance function of Wang and Zhou(2018b)under this tax structure.A sufficient condition is given under which a unique level for starting taxation exists.The results considerably generalise theorem 3 of Albrecher and Hipp(2007).