New directions in insurance loss modeling, ruin theory and their applications

本研究项目继续寻求数学风险理论的新的理论和实践成果，将该领域与其他保险或金融领域联系起来，如保险损失建模、索赔准备金和操作风险管理。当两家或更多的保险公司同时参与时，考虑的是现实情况。再保险人弥补保险人部分损失的情况，或两家或两家以上保险人试图确定将某些风险措施降至最低的最佳风险分担策略的情况，将详细探讨。已发生但未报告的损失准备金和已报告但未结算的准备金是索赔准备金中的重要数量，并将通过其经典的破产相关指标-破产时间、破产前盈余和破产时赤字-纳入保险公司的盈余分析。在更实际的方面，对保险损失进行建模是精算师面临的最具挑战性的问题之一。这项建议要解决的一个重要问题是确定潜在的候选者，在处理截断和审查的数据集时，可以用这些候选者来拟合损失量。由于其非常吸引人的性质(稠密性、封闭性等)，混合Erlang分布类将在基于著名的EM方法的几种拟合算法中发挥主要作用。将详细研究所提出的基于Erlang的方法在索赔准备中的潜在应用。金融机构面临着战略风险、声誉风险、市场风险、信用风险、业务风险、操作风险、流动性风险等多种风险。在这些风险中，市场风险、信用风险和操作风险在实践中很容易量化。在这个项目中，我们感兴趣的是根据巴塞尔II指令中提出的高级测量方法(AMA)来衡量操作风险。为了避免模拟，我们有兴趣找到与操作风险相关的总体损失分布的封闭形式的解析解，并考虑到不同测量单位之间可能出现的某些相关性结构。

This research project continues the search for new theoretical and practical results in mathematical risk theory, connecting the area with other insurance or financial areas, such as insurance loss modeling, claim reserving and operational risk management. Realistic scenarios are considered when two or more insurance companies are at play at the same time. Situations where a reinsurer covers parts of the insurer losses, or where two or more insurers try to determine an optimal risk-sharing strategy that minimizes certain risks measures, will be pursued in details. The incurred but not reported loss reserve, and the reported but not settled reserve are important quantities in claim reserving, and will be included in the analysis of the surplus of an insurance company via its classical ruin related measures - the time to ruin, the surplus prior to ruin and the deficit at ruin. On a more practical side, modeling insurance losses is one of the most challenging problems for actuaries. An important question to be addressed in this proposal is the identification of potential candidates which can be used for fitting loss amounts when dealing with truncated and censored data sets. Due to its very appealing properties (denseness, closure etc.), the mixed Erlang class of distributions will play the main role in developing several fitting algorithms based on the well-known EM method. Potential applications of the proposed Erlang based methods in claim reserving will be studied in detail. Financial institutions are exposed to a variety of risks such as, strategy risk, reputation risk, market risk, credit risk, business risk, operational risk, liquid risk, and so on. Among these risks, market, credit and operational risks can be easily quantified in practice. In this project we are interested in measuring the operational risk according to the Advanced Measurement Approach (AMA) proposed in the Basel II directive. In order to avoid simulation, we are interested in finding closed form analytic solutions for the distribution of the overall losses associated to the operational risk, taking into consideration certain dependence structures that may occur among the different units of measurement.