Draw-down Based Two-sided Exit Problems For Spectrally Negative Lévy Processes Observed At Poisson Arrival Times

Two-sided exit problems for spectrally negative Lévy processes have been studied extensively over the past ten years,which find many applications in risk theory,queues and continuous branching processes.Recently,based on theories of Kuznetsov(2006)and Albrecher and Ivanovs(2013),Albrecher et al.(2016)extended various two-sided exit identities to the situation when a spectrally one-sided Lévy process is only observed at arrival epochs of an independent Poisson process.Quite recently,by using a very delicate approximating method and excursion theory,Li et al.(2017)found expressions of Laplace transforms for the two-sided exit problems involving the draw-down time.Motivated by Albrecher et al.(2016)and Li et al.(2017),the present paper concerns draw-down based two-sided exit problems for spectrally negative Lévy processes observed at Poisson arrival times.Our problem is separated into three sub-problems:two-sided ex-it problems for continuous and Poisson observations,two-sided exit problems without the draw-down time and problems of joint occupation times.By using Poisson approach of Li and Zhou(2014),properties of scale functions,the strong Markov property and a modified approximating method of Li et al.(2017)together with the compensation formula in ex-cursion theory,we find expressions of Laplace transforms for the two-sided exit problems involving the draw-down time when a spectrally negative Lévy process is only observed at arrival epochs of an independent Poisson process,which extends results of Albrecher et al.(2016).Furthermore,we simplify our results in the special case.As an example,the Cramer-Lundberg risk model is provided at the end of this paper to study draw-up time.By using the theory of the Gerber-Shiu measure mentioned in Kyprianou(2013),differential equations which draw-up probability and Laplace transforms involving the draw-up time satisfy are derived.Therefore,we find draw-up probability and expressions of Laplace transforms involving the draw-up time.