一类基于整数值时间序列的风险模型分析

Risk theory is one of the most active research areas of actuarial science. Under some practical conditions, we can model the surplus process of the insurance company, analysis and measure the risk quantitatively. In the classical risk model, the claim numbers of different periods are supposed to be stationary and independent. However, it is not true in practice. In our project, we will use integer-valued time series based on signed thinning operator (such as INAR, INMA, etc.) to describe the claim numbers of insurance companies and show the nonstationarity and negative correlation of the real data, and based on these integer-valued time series, we propose a new kind of risk model in which the claim numbers of different periods are dependent. Firstly, we discuss some probabilistic and statistical properties of the new models. Secondly, we study the calculation method of related risk measures (ruin probability, Gerber-Shiu function, VaR, CVaR, etc.) in detail, and extend them to the multidimensional risk model. At last, we consider how to control risks by investment, reinsurance and dividend. The project involves three research fields: finance, actuarial science and time series, which are all hot topics among the international academy. The new research results of this project will enrich the content of related subjects, expand the actual application of time series, and provide more effective theoretic guide for insurers to develop new business and manage their risks.

风险理论是精算学中最活跃的研究领域之一，它的研究方法是通过建立保险公司资本流动情况的盈余过程，对其风险进行定量分析和管理。传统的风险模型一般假设不同时期的理赔次数平稳独立，而在实际中，这通常不成立。本项目中，我们用基于符号thinning算子的整数值时间序列模型(如整数值AR、MA等)来描述理赔次数，反映实际数据的非平稳和负相关性，并基于此建立不同时期的理赔次数具有一定相依关系的风险模型。首先，我们讨论新模型的一些概率统计性质；其次，着重研究其风险测度(破产概率、Gerber-Shiu函数、VaR、CVaR)的计算方法，并把相应结果推广到多维风险模型的情形；最后，考虑投资、再保险、分红策略等风险管理问题。本项目涉及金融、精算和时间序列这三个国际热门的研究领域，研究结果将丰富这些研究领域的理论内容，拓展时间序列在实际中的应用，为保险经营者和决策者发展业务、进行风险管理提供更有效的理论指导。

本项目意在讨论几类新的基于统计模型的盈余过程，并考虑相应的风险度量和风险管理问题。首先，建立了四类新的风险模型，并给出了相应的概率统计性质。其次，分析了新模型的风险度量问题，得到了破产概率的计算和估计方法，并利用数值模拟验证了新模型的优越性。最后，讨论了可加风险模型和部分线性单指标模型的参数估计问题，为研究成果在实务中的应用奠定了基础。