Levy processes and related problems

Levy过程是具有独立平稳增量的随机过程。最著名的例子是布朗运动和复合泊松过程。对于这类过程，已经发展了丰富的数学理论，并在排队论、保险风险理论、数理金融学以及连续状态分支过程的谱系结构等方面得到了广泛的应用。本项目的一个目的是研究Levy过程及其相关过程的极值性质。我们感兴趣的问题之一是理解这样一个过程是如何从一个区间中首先退出的。我们还对这些过程的范围时间、当地时间和漂移感兴趣。作为一个应用，上述工作使我们能够引入一个新的风险模型，我们可以动态地调整其保费费率。该模型是经典风险模型和带障碍风险模型之间的折衷。它更现实，同时在数学上仍然容易处理。得到了破产问题的一些显式结果。另一个目标是探讨几个测度值随机过程，其中Levy过程是深入参与。这类过程包括Levy分支超过程、Levy空间运动合并超过程和Levy迁移的垫脚石模型。这个项目将有助于随机过程的波动理论，特别是Levy过程。它将为风险理论的研究带来新的模型和新的技术，如漂移理论。这也将导致更好地理解和可能的解决方案，一些具有挑战性的问题上的测度值过程。

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.