The Last Exit Time For Spectrally Negative Lévy Processes And Its Related Problems

Occupation time usually means the sum of the time that a random process stays in a specific area,and its results are widely used in the research of mathematical finance and risk theory.Recently,the last exit time also gradually attracted attention,the last exit time generally indicates the time when the final departure time is from a specific area,it is regarded as the final bankruptcy time in the risk model,that is to say,there is no bankruptcy after that,and the process remains in the negative half axis.The last exit time is of great significance for the study of risk model to get rid of negative surplus and complete bankruptcy.This paper mainly uses Poisson method and perturbation method are used to combine some exit identities of spectral negative Lévy processes,process space homo-geneity and strong Markov property,in which the perturbation method mainly solves the case of infinite motion of the process,and the Poisson method mainly solves the case of unbounded variation path of the process,study the joint Laplace transform of the positive half axis last exit time and occupation time and the joint Laplace trans-form of the positive half axis last exit time,occupation time and value of the process at the last time.