The Occupation Time And Resolvents Of Spectrally Negative Lévy Processes Related To The Last Exit Time

In recent years,many scholars at home and abroad have begun to study the hot issues related to the last exit time.In this paper,the Possion method and the perturbation method in Li et al(2017)are used to study the Laplace transform of the first exit time and its occupation time for the spectrally negative Levy process related to the last exit time.Using the Wiener-Hopf factorization of Kuznetsov et al(2013),the resolvents of the spectrally negative Levy process related to the last exit time are calculated.The full text is divided into four chapters.The first chapter mainly introduces the research background and present situation of the occupation time and the resolvents related to the last exit time,and expounds the main results and innovations of this paper.In the second chapter,we introduce some basic knowledge about the spectrally negative Levy process and the related fluctuation identities.In chapter 3,Possion method and perturbation method are used to calculate the joint Lalace transform of the first exit time and the associated occupation time for spectrally negative Levy processes related to the last exit time,i.e for any ?,p>0 and a>0,x?a,for any ?,p>0 and a>0,x?-a,And T0+=sup{t? 0:Xt<0},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 the fourth part,by using the Wiener-Hopf factorization,we consider some resolvents for spectrally negative Levy processes on the last exit time,such as for arbitrary Borel set A ?[0,?),for arbitrary Borel set A?[0,a],...