The Laplace Transforms For Spectrally Negative Lévy Processes At The Last Time Of Exiting The Negative Part Of Real Line

Occupation time is a hot issue in theoretical studies of stochastic processes,its results are widely used in mathematical finance and insurance risk models.In recent years,the last exit time has also aroused widespread concern of scholars at home and abroad.In a risk model,it can be regarded as the final recovery time after when there will be no more ruin.The main purpose of this paper is to study the laplace transforms for spectrally negative Lévy processes at the last time of exiting the negative part of real line.To overcome difficulties due to the non-stoping last time and the infinite activity of the Lévy processes,we use the Poisson approach to find Laplace transforms of the occupation time and apply a perturbation approach to evaluate the last time of exiting the negative part of real line by first exit times over certain events.This paper is divided into four chapters,arranged as follows:In chapter 1,we introduce the research background and current situation,and then outline the main contents and innovation of this paper.In chapter 2,we introduce spectrally negative Lévy processes,scale functions and some fluctuation identities.In chapter 3,we use a perturbation approach to find exit identities between the last time of exiting the negative part of real line and the first exit time for spectrally negative Lévy processes,such as(?)and(?)at the end of this chapter,our results obtained are verified.And T0+=sup{t?0:Xt<0},?a-=inf{t>0:Xt<a},?a+=inf{t>0:Xt>a},with the convention sup(?)=0,inf(?)=?.In chapter 4,using the Poisson approach and the perturbation approach,we consider the joint laplace transforms for spectrally negative Lévy processes at the last time of exiting the negative part of real line and occupation time,such as(?)and(?)At the end of this chapter,our results obtained are verified.