Insurance risk analysis with dependence and reinsurance

经典风险理论通常研究一家公司或某一行业的准备金过程。然而，一家公司可以经营几个业务线，也可以有几个子公司。这些业务的储备过程形成了一个多变量的相关风险过程。即使对于不同的公司，由于一些共同的经济和金融因素，储备过程也可能是相关的。此外，当一家保险公司与一家再保险公司签订再保险协议时，两家公司的准备金风险过程也是相关的。因此，相关性自然而然地出现在多变量准备金风险过程和再保险研究中。 在这个项目中，我们将重点研究二维准备金风险过程。对于二维准备金风险模型中的每一项业务，我们将违约时间定义为该业务的准备金首次跌破某一水平的时间。我们将研究不同类型违约的概率。我们还将研究与每项业务的索赔相关的预期总贴现费用，直至违约次数。这种预期的总贴现费用可以用来研究许多与违约相关的数量，并确定每一项业务的资本要求。我们假设再保险策略适用于一项业务。我们将研究不同优化准则下相关业务的最优再保险策略。进一步，我们将考虑二维准备金风险过程中的经济和金融因素，如利率和股息，并讨论经济和金融因素对违约概率的影响。该项目将为具有相关性和再保险的保险风险分析提供新的随机模型，并揭示相关性、再保险以及一些经济和金融因素对违约次数和违约概率的影响。预期结果不仅可用于保险风险分析，还可用于信用风险分析等应用概率领域。

Classical risk theory usually studies a reserve process for a company or a line of business. However, a company could run several lines of business or have several subcompanies. The reserve processes of these lines of business form a multivariate dependent risk process. Even for different companies, the reserve processes could be correlated due to some common economic and financial factors. Furthermore, when an insurance company has a reinsurance treaty with a reinsurance company, the reserve risk processes of the two companies are correlated as well. Thus, dependence naturally appears in a multivariate reserve risk process and in the study of reinsurance. In this project, we will focus on two-dimensional reserve risk processes. For each line of business in a two-dimensional reserve risk model, we define default time as the first time when the reserve of the line falls below a certain level. We will study the probabilities of different types of defaults. We will also study the expected total discounted expenses related to claims of each line of business up to default times. Such expected total discounted expenses can be used to study many default-related quantities and to determine the capital requirements for each line of business. We assume that a reinsurance strategy is applied to one line of business. We will study the optimal reinsurance strategies for dependent lines of business under different optimization criteria. Further, we will consider economic and financial factors, such as interest rates and dividends, in two-dimensional reserve risk processes and discuss the influence of economic and financial factors on default probabilities. The project will provide new stochastic models for insurance risk analysis with dependence and reinsurance, and reveal the impact of dependence, reinsurance, and some economic and financial factors on default times and default probabilities. The anticipated results will be useful not only in insurance risk analysis but also in credit risk analysis and other fields of applied probability.