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。

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.