Risk models based on Marked Markovian Arrival Processes

保险公司通常面临多种来源（类型）的损失。因此，对不同风险来源之间的相互依赖关系进行建模是极其重要的。文献中提出了一些多变量风险模型。然而，大多数模型要么关注索赔数量之间的依赖关系，要么关注索赔规模之间的依赖关系。在本研究计划中，我们提出了一个框架，用于建模索赔数量之间的相互依赖关系，索赔规模之间，以及索赔数量和索赔规模之间。我们将研究模型的各个方面：参数估计、风险度量、破产概率和扩展。在该计划中开发的方法将是排队理论，统计学和精算科学的结合。它们可以应用于其他领域的应用程序。我们提出的模型是基于标记马尔可夫到达过程（MMAP），该过程是由He和Neuts（随机过程及其应用，1998）在排队理论文献中引入的。MMAP是马尔可夫到达过程（Markov Arrival Processes， MAP）的推广，广泛应用于互联网流量建模等领域。当用于保险损失建模时，MMAP结构允许索赔数之间、索赔规模之间以及索赔数与索赔规模之间的依赖关系。关于MMAP风险模型已经做了一些工作。例如，在Ren （Insurance, Mathematics and Economics, 2012）中，我们提供了各种类型损失的联合力矩和联合分布的公式。在Ren （Stochastic Models 2013）中，我们分析了由于不同类型的损失而导致破产的概率。在本研究计划中，我们计划实施以下项目：1)尽管MMAP的概念在排队文献、通信系统和其他应用领域得到广泛接受，但其参数估计的统计方法尚未得到广泛研究。我们建议在模型建立的基础上研究MMAP的参数估计方法。所开发的估计方法一方面将促进MMAP在精算科学中的应用；另一方面，它可以促进MMAP在其他领域的应用。2)已发生但未报告的索赔金额（IBNR）索赔是保险公司的财务责任。估计IBNR对他们来说非常重要。Willmot（精算研究结算所，1990）通过假设索赔根据泊松（混合泊松）过程到达来分析IBNR问题。在这个项目中，我们建议通过假设有多种类型的保险索赔并且索赔根据MMAP到达来分析IBNR问题。这将大大推广以前的结果。3)多元风险过程中破产概率的分析是非常困难的。然而，MMAP结构允许直接模拟多元破产概率的方法。在这个项目中，我们将尝试研究是否可以使用诸如重要性采样之类的技术来提高模拟的速度和/或效率。4)扩展MMAP风险模型，使其包含索赔强度和索赔规模分布的解释变量。显然，具有某些特征的投保人更容易遭受损失(例如：年轻的男性司机)会招致更严重的损失。在模型中加入解释变量可能有助于将保单持有人分类为更均匀的群体，然后可以相应地收取保险费。该项目与财产险和意外险中使用的流行预测模型高度相关。

Insurance companies typically face multiple sources (types) of losses. Therefore, it is extremely important to model the inter-dependencies among the different sources of risk. There are some multivariate risk models proposed in the literature. However, most of the models either focus on the dependency among the number of claims or among the sizes of claims. In this research program, we propose a framework for modelling the inter-dependencies among the number of claims, among the sizes of claims, as well as between the claim numbers and claim sizes. We will investigate various aspects of the model: parameter estimations, measures of risk, ruin probabilities and extensions. The methodology developed in the program will be a combination of queueing theory, statistics, and actuarial science. They can be applied to other areas of applications. The model that we propose is based on the Marked Markovian Arrival Processes (MMAP), which were introduced by He and Neuts (Stochastic Processes and their Applications ,1998) in queueing theory literature. MMAP generalizes the well known Markov Arrival Processes (MAP), it is widely used in internet traffic modeling and other areas. When used for modelling insurance losses, the MMAP structure allows dependencies among claim numbers, among claim sizes and between claim numbers and claim sizes. Some work has been done with regard to the MMAP risk model. For example, in Ren (Insurance, Mathematics and Economics, 2012), we provided formulas for the joint moments and joint distribution of various types of losses. In Ren (Stochastic Models 2013), we analyzed the probabilities of ruin due to different type of losses.In this proposed research program, we plan to implement the following projects:1) Although the concept of MMAP has been widely accepted in queueing literature, in communication systems, and other application areas, the statistical methods of parameter estimation for it has not been studied extensively. We propose to study the parameter estimation methods for MMAP based on our model setups. The estimation methods developed on the one hand will facilitate the application of MMAP in actuarial science; on the other hand, it may promote applications of MMAP to other areas.2) The amount of claims that have incurred but not reported (IBNR) claims is a financial liability to an insurance company. Estimating IBNR is extremely important for them. Willmot (Actuarial Research Clearing House, 1990) analyzed the IBNR problem by assuming that the claim arrives according to a Poisson (mixed Poisson) process. In this project, we propose to analyze the IBNR problem by assuming that there are multiple types of insurance claims and the claims arrive according to a MMAP. This will significantly generalize the previous results.3) It is known that the analysis of ruin probabilities in multivariate risk processes is very hard. However, the MMAP structure allows a straight forward approach of simulating multivariate ruin probabilities. In this project, we will try to investigate whether techniques such as importance sampling could be used to improve the speed and/or efficiency of the simulation.4) Extending the MMAP risk model to include explanatory variables for both claim intensities and claim size distributions. Evidently, policyholders with certain characteristics are more likely to incur losses (eg. Young male drivers) and incur more severe losses. Including explanatory variables in the model may help classifying policy holders into more homogeneous groups and the insurance premiums then can be charged accordingly. This project is highly relevant to the popular predictive modelling used in property and casualty insurance.