几类含借贷利率风险过程的绝对破产与分红问题

The project mainly studies absolute ruin and dividend problems for some kinds of risk processes with debit interest. This is the application of stochastic process theory in the field of insurance and is a new research direction in risk theory. In order to describe the actual situation that the insurance company can borrow money to continue the operating, we will study the risk processes with debit interest, and the corresponding bankruptcy problem is often referred to as the absolute ruin problem. In this project, we will study the following problems: one, absolute ruin theory for a kind of Cox risk process with debit interest and the perturbed Cox risk process with debit interest, which involves the actuarial diagnostics including Gerber-Shiu function, the hitting time, the last exit time and so on. At the same time we will also study the precise asymptotic formula for absolute ruin probability when the claim distribution is heavy tailed. Two, dividend problems for a class of Levy risk process with debit interest. We will study the expected discounted dividend function expression and differentiability under different dividend strategies and find the conditions that the optimal dividend strategy is barrier or threshold strategy. It is the current trend to apply the properties of Cox process and Levy process into the study of risk theory. The study of these problems in this project not only conforms to the development of the insurance industry, but also has active sense in enriching the stochastic process theory.

本项目主要研究几类含借贷利率风险过程的绝对破产与分红问题，这是随机过程理论在保险领域的应用，属于风险理论中一个新的研究方向。为描述保险公司可以借贷以继续运营这一实际情况，我们将风险模型刻画为含借贷利率的风险过程，相应的破产问题常称为绝对破产问题。本项目中我们将研究如下问题：一、含借贷利率的一类Cox风险过程以及带扰动Cox风险过程的绝对破产理论，其中所涉及的精算量包括Gerber-Shiu函数、首中时、末离时等。同时我们还将研究索赔分布为重尾情形下绝对破产概率的精确渐进式。二、含借贷利率的一类Levy风险过程的分红问题。我们将研究不同分红策略下期望折现分红函数的表达式以及可微性，同时寻求使得边界分红或者门槛分红策略为最优分红策略的条件。将Cox过程和Levy过程的性质应用到风险理论中去是当前的研究趋势。本项目研究问题的解决不仅符合保险事业的发展，同时对于随机过程理论的丰富也具有积极意义。

本项目主要研究几类含借贷利率风险过程的绝对破产与分红问题，我们在该领域的研究中取得了一些进展。目前已有1篇论文正式发表，2篇论文已经接收并在线刊出，等待正式发表。我们考察了一类含借贷利率风险过程的负持续时间问题，计算出总的负持续时间的拉普拉斯变换。我们研究了一类含常利率的带扰动更新过程，得到了在门槛分红策略下该风险过程的期望折现分红量。特别的在索赔间隔服从Erlang(2)时，我们用数值例子解释了所得结果。这些结果均以论文形式进行了发表。同时，对于几类模型的最优分红问题的结果，我们正在整理论文。本项目初步完成了大部分预期结果。