几类风险过程的实质性破产问题

The project mianly studies bankruptcy problems for some kinds of risk processes. This is a new research direction of application of stochastic process theory in risk theory. The concept of bankruptcy is proposed recently to reflect the actual situaion that when the surplus is negative, the insurance company can do business as usual with some probability until bankruptcy occurs. In this project, we will study the following problems: one, bankruptcy problems for perturbed classical risk process and Levy risk process, which involves the acturial diagnostics bankruptcy probability, Gerber-Shiu function at bankruptcy and so on. At the same time we will also study the asymptotic formula for bankruptcy probability when the claim distribution is heavy tailed. Two, bankruptcy probblems for Markov-modulated risk processes. Markov-modulated compound Poisson process and Markov-modulated diffusion process are the main concerned risk processes. We will study the duration of negative surplus except bankruptcy probability. Three, barrier dividend problems for diffusion risk process with stochastic return on investment at bankruptcy. We will compute the expexted discounted dividends until bankruptcy and give the optimal barrier. The study of these problems in this project not only conforms to the development of the insurance industry, but also has active sense in studying the stochastic process theory.

本项目主要研究几类风险过程的实质性破产问题，这是随机过程理论在风险理论中应用的一个新的研究方向。实质性破产的概念是近几年提出来的，用来反映保险公司盈余过程达到负值时并不立即破产，而是以某概率继续运营这一实际情况。本项目将要研究的问题为:一、带扰动古典风险过程及更一般的Levy风险过程的实质性破产问题。其中所涉及的精算量主要是实质性破产概率和实质性破产时刻的Gerber-Shiu期望折现罚金函数。同时还将考虑索赔分布为重尾情形下实质性破产概率的渐近表达式。二、马氏调节风险过程的实质性破产问题。其中主要涉及的是马氏调节复合泊松过程和马氏调节扩散过程。除实质性破产概率外，重点研究负持续时问题。三、含随机投资回报扩散风险过程实质性破产条件下的边界分红问题。我们将计算实质性破产时刻前的期望折现分红量，并给出最优边界。本项目中问题的解决符合当前保险事业的发展，对随机过程的理论研究也具有积极意义。