Levy processes and related problems

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Levy processes are stochastic processes with independent and stationary increments. The best known examples are Brownian motion and the compound Poisson processes. For such processes, a rich mathematics theory has been developed, and extensive applications have been found in queuing theory, risk theory for insurance, mathematical finance and the genealogical structures of continuous state branching processes. One objective of this project is to study the properties concerning the extremes of Levy processes and related processes. One of the questions we are interested in is to understand how such a process first exits from an interval. We are also interested in the range time, the local time, and the excursions for these processes. As an application, the above-mentioned work enables us to introduce a novel risk model in which we can dynamically adjust its premium rate. This model is a compromise between the classical risk model and the risk model with barrier. It is more realistic while still mathematically tractable. Some explicit results on the ruin problem can be obtained. The other objective is to explore several measure-valued stochastic processes in which Levy processes are deeply involved. Such processes include the superprocess with Levy branching, the superprocess with coalescing Levy spatial motion, and the stepping-stone model undergoing levy migration. This project will contribute to the fluctuation theory for stochastic processes, in particular, for Levy processes. It will bring in new models and new techniques, such as excursion theory, to the study of risk theory. It will also lead to a better understanding and possible solutions to some challenging problems on measure-valued processes.