多维风险模型及其若干相关问题研究

Multidimensional risk theory represents a very attractive topic that has gained a lot of popularity in recent few years. The main goals of this project are to study multidimensional(multivariate) risk models and to further explore some of the known results (of the applicant or other researchers) on univariate risk models. The target questions in this project are based on the very recent exiting developments in the area of univariate or multivariate risk models, and they can be divided into the following four directions..　　1. For an insurance company with several business lines, we denote by a multidimensional compound Poisson risk model it’s surplus process. Some very recent literature have been devoted to maximizing the company’s joint survival probabilities by way of improving their internal risk models or using reinsurance. However, the allocation of the total initial capital held by an insurance company with different business lines has not been considered as a way of survival improvement, which is subject to further study in this project. That is we will choose, among the numerous ways of allocating the aggregate initial capital to its different business lines, the one that makes improvement to or maximizes the joint and individual survival probability..　　2. The applicant will continue the study of problem 1, but this time the multidimensional compound Poisson risk process is replaced with the multidimensional Lévy risk process. Although moving to this more complex setting does not bring any more realistic feature, a clear mathematical advantage has emerged. For example, researchers tend to prefer to address the ruin problem via the fluctuation theory..　　3. To the best of the applicant's knowledge, the optimal dividend strategy in the multidimensional insurance risk model is still unknown, although it is already clear in the univariate risk model. The applicant wants to explore, among all admissible dividend distribution strategies, the most favorable one which gives out the maximum expected discounted dividend payments..　　4. Albrecher and Hipp (2007) first introduced the so-called loss-carry-forward taxation into the classical(univariate) compound Poisson risk model. Since then a lot of interesting results concerning the ruin probability, expected discounted taxation and the Gerber-Shiu function have emerged. The applicant wants to continue the study in multidimensional insurance risk model. In words, we will investigate how tax influences the qualitative and quantitative behavior of the infinite time ruin probability, and the problem of optimal taxation strategy is also a target question in this project.

本项目拟研究多维保险风险模型，并对一些一维风险模型中的现有结果进行更深入的探索。其中拟解决的问题都是基于最近几年来申请者和其他研究者在一维或多维风险模型中取得的一些新的研究进展，具体分为以下四个方面：. 1，对于一个拥有多条经营线(business lines)的保险公司，以多维复合泊松模型模拟其盈余过程，优化保险公司初始资本的分配方式(将保险公司的总初始资本分配到各条经营线)，使联合生存概率和边际生存概率更大(或最大)；. 2，在多维谱负列维过程的框架下继续探讨问题1；. 3，基于一维风险模型中最优分红问题的已有结果，继续在多维风险模型中讨论(最优)分红问题，探索最优分红策略；. 4，在多维风险模型中讨论“loss-carry-forward”赋税问题，定性定量地研究“loss-carry-forward”税是如何影响破产概率的，并探索最优赋税策略。

本项目研究一维或多维保险风险模型，其中解决的问题都是基于最近几年来申请者和其他研究者在一维或多维风险模型中取得的一些新的研究进展：.1、以控制保险公司的风险为研究动机，研究最优再保险问题（考虑一次性再保险优化，也考虑再保险策略的长期动态优化，共产出科研论文5篇，发表4篇）。.2、以优化保险公司的红利分配策略为动机，研究最优分红问题（共产出科研论文3篇，发表3篇）。.3、以优化保险公司赋税策略为动机，研究最优赋税问题（共产出科研论文4篇，发表4篇）。.4、已优化保险公司的生存概率以改善公司的经营状况为动机，研究具有多条经营线的保险公司的经营状况优化问题（产出论文1篇，已投稿）。