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极其重要。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相关性，要么关注索赔规模之间的相关性。在这个研究项目中，我们提出了一个框架，用于建模索赔数量之间、索赔规模之间以及索赔数量和索赔规模之间的相互依赖关系。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。在该项目中开发的方法将是排队论、统计学和精算学的结合。它们可以应用于其他应用领域。 我们提出的模型是基于由He和Neuts(随机过程及其应用，1998)在排队论文献中提出的标记马尔可夫到达过程(MMAP)。MMAP是著名的马尔可夫到达过程(MAP)的推广，广泛应用于网络流量建模等领域。当用于对保险损失进行建模时，MMAP结构允许索赔数量之间、索赔金额之间以及索赔数量和索赔金额之间的依赖关系。关于MMAP风险模型已经做了一些工作。例如，在REN(保险、数学与经济，2012)中，我们提供了各种类型损失的联合矩和联合分布公式。在REN(随机模型2013)中，我们分析了由于不同类型的损失而导致的破产概率。 在这项拟议的研究计划中，我们计划实施以下项目： 1)虽然MMAP的概念在排队学文献、通信系统和其他应用领域中已经被广泛接受，但它的参数估计的统计方法还没有得到广泛的研究。基于我们建立的模型，我们建议研究MMAP的参数估计方法。所开发的估计方法一方面有利于MMAP在精算领域的应用，另一方面也可能促进MMAP在其他领域的应用。 2)已发生但未报告的索赔金额(IBNR)是保险公司的财务负债。对他们来说，估计IBNR是非常重要的。Willmot(Actuariary Research Clearing House，1990)通过假设索赔是按照泊松(Mixed Poisson)过程到达的，分析了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。这将极大地推广先前的结果。 3)众所周知，多元风险过程中的破产概率分析是非常困难的。然而，MMAP结构允许一种直接的方法来模拟多变量破产概率。在这个项目中，我们将尝试调查是否可以使用重要性抽样等技术来提高模拟的速度和/或效率。 4)扩展了MMAP风险模型，将索赔强度和索赔规模分布的解释变量都包括在内。显然，具有某些特征的投保人更有可能蒙受损失(例如，年轻男性司机)，并招致更严重的损失。在模型中加入解释变量可能有助于将投保人归入更同质的群体，然后可以相应地收取保险费。该项目与财产和意外伤害保险中流行的预测模型高度相关。

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.