带控制变化尾相依随机变量和的偏差定理及相关问题

This project focuses on the study of deviation theorems for sums of dependent random variables with dominatedly varying tails. Our main task include the following aspects: (1) large and moderate deviation theorems for non-random and random sums of END random variables with dominately varying tails in single-risk models; (2)large and moderate deviation theorems for partial and random sums of ENA random arrays with dominately varying tails in multi-risk models. All the problems mentioned above have good backgroud and are hot spots of study on insurance and actuarial science around the world. To use many mathematical tools such as probablity theory、stochatic process、probability limit theory efficiently is the key to solve these problems. We anticpate the results can offer theoretical help for financial institution、 insurer and inspecting agency to deal with risk management and investment decision-making. This project can also promote the cross of mathematics and financial science. Moreover, it can enrich and develop the theory of financial mathematics.

本项目旨在研究带控制变化尾相依随机变量和的偏差定理，主要包括：（1）单风险模型中，带控制变化尾END随机变量确定和与随机和的大偏差与中偏差定理。（2）多风险模型中，带控制变化尾ENA随机阵列部分和与随机和的大偏差与中偏差定理。以上所涉问题金融保险背景鲜明，均属当前国际上保险精算学研究的热点问题。概率论、随机过程、概率极限理论等多种数学工具的有效使用是解决问题的关键。预期成果可为金融保险机构及监管部门的风险管理和投资决策提供可靠的理论依据，促进数学学科与金融学科的交叉，丰富和发展金融数学理论。

本项目旨在研究带控制变化尾相依随机变量和的偏差定理。得到了END相依结构下实值一致变化随机变量随机和的一般情形下的精细大偏差定理，研究了ENA相依结构下带一致变化尾相依多风险模型的偏差定理，给出了带控制变化尾LENOD相依结构下多风险模型的偏差定理, 目前尚有部分与本项目有关的结果正处于整理过程中。所获得的的研究成果较好的达到了预期研究目标。