网络马氏骨架过程框架下的保险风险研究
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摘要
网络马氏骨架过程是马志明院士的科研团队近期提出的一类新过程,这是一个包含马氏过程在内的范围很广的过程类.这类过程非常适合模拟随机变量之间的回归型相依关系.本论文进一步研究网络马氏骨架过程,并讨论其在保险风险中的应用.首先引入复合网络马氏骨架过程的概念,并讨论其极限性质.其次重点研究网络马氏骨架框架下总索赔量的精细大偏差和破产概率等问题.
直观上来说,一个网络马氏骨架过程(web Markov skeleton process,简记为WMSP)是一个纯跳的马氏骨架过程,并且跳间隔时间序列在给定其骨架信息的条件下是相互独立的，一般地,一个WMSP的动态机制可以描述为如下形式：其中{Xn,n≥0}是马尔科夫链,{Tn,n≥0}是跳间隔时间序列.
我们引入复合网络马氏骨架过程(compound web Markov skeleton process,简记为CWMSP)它的机制可以描述如下：这里Nt表示到时间t为止跳跃的次数,×是伴随第i次跳跃的随机变量.
由CWMSP描述的机制出现在许多自然和社会科学学科中,比如生物学,金融,排队论和保险等等,这里我们关注的是其在保险中的应用.风险的相依性是保险理论的研究热点,也是难点.相对于更新风险模型及其一般的拓广模型,在保险理论中运用网络马氏骨架过程来建模能够充分的考虑历史信息的相依性问题,这大大拓展了模型的适用性.
论文主要包含五个部分：第一部分是绪论和准备知识部分,由第一章构成,第二部分介绍网络马氏骨架过程并引入复合网络马氏骨架过程概念,给出了特定条件下复合网络马氏骨架过程的精细大偏差公式,这部分由第二章和第三章构成.第三部分即第四章,给出了特定条件下总索赔量的精细大偏差公式.第五章到第七章,给出了一个特殊的复合网络马氏骨架过程,主要讨论特定重灾风险模型下的破产问题,包括有限时间破产概率,无限时间破产概率及破产概率的局部结果等问题.最后是总结和展望.
Web Markov skeleton process (WMSP for short) is a new class of processes re-cently introduced by Ma's group. Based on the work of Ma's group, it is the first time this thesis introduces the notation of compound web Markov skeleton process (CWMSP for short), then it studies some insurance risk problems under the framework of web Markov skeleton process.
As a hot area of research in risk theory, dependent risk is our focus in this paper. As a new class of processes, it applicable to model the dependence in risk theory, which will be a breakthrough in this area.
Intuitively, a Markov skeleton process is a stochastic process employing a Markov chain as its skeleton. A web Markov skeleton process is a jump process and also a Markov skeleton process in which given information of its skeleton, the time slots between jumps are conditionally independent to each other. The system of a WMSP can be described as follows: where{Xn, n≥0} is a Markov chain, and{Tn, n≥0} is the set of time slots between adjacent jumps.
A compound web Markov skeleton process (CWMSP for short) can be written as where Nt is the associated counting process, which denotes the number of jumps before t, and Yi is the accompaniment of ith jump, which is a r.v.. The system of a CWMSP can be described as follows:
The system described by CWMSPs appear in various natural and social sciences, such as biology, finance, queueing theory, insurance and so on. In particular, CWMSP is a suitable framework for modeling the aggregate amount of claims when the claim sizes and inter-claim times are not independent.
This thesis has five parts:The first part is the introduction and preliminaries, which are the main content of Chapter1. In the second part we introduce WMSP and CWMSP, and give the precise large deviation for CWMSP under some condition. And this part consists of Chapter2and3. The third part is Chapter4, which gives the precise large deviation for the aggregate amount of claims under some specified condition. The main topic of Chapter5,6and7is the discussion of the ruin problem under a given catastrophe risk model, which includes finite-time ruin probability and infinite time ruin probability. The last part is the conclusion and the future work.
直观上来说,一个网络马氏骨架过程(web Markov skeleton process,简记为WMSP)是一个纯跳的马氏骨架过程,并且跳间隔时间序列在给定其骨架信息的条件下是相互独立的，一般地,一个WMSP的动态机制可以描述为如下形式：其中{Xn,n≥0}是马尔科夫链,{Tn,n≥0}是跳间隔时间序列.
我们引入复合网络马氏骨架过程(compound web Markov skeleton process,简记为CWMSP)它的机制可以描述如下：这里Nt表示到时间t为止跳跃的次数,×是伴随第i次跳跃的随机变量.
由CWMSP描述的机制出现在许多自然和社会科学学科中,比如生物学,金融,排队论和保险等等,这里我们关注的是其在保险中的应用.风险的相依性是保险理论的研究热点,也是难点.相对于更新风险模型及其一般的拓广模型,在保险理论中运用网络马氏骨架过程来建模能够充分的考虑历史信息的相依性问题,这大大拓展了模型的适用性.
论文主要包含五个部分：第一部分是绪论和准备知识部分,由第一章构成,第二部分介绍网络马氏骨架过程并引入复合网络马氏骨架过程概念,给出了特定条件下复合网络马氏骨架过程的精细大偏差公式,这部分由第二章和第三章构成.第三部分即第四章,给出了特定条件下总索赔量的精细大偏差公式.第五章到第七章,给出了一个特殊的复合网络马氏骨架过程,主要讨论特定重灾风险模型下的破产问题,包括有限时间破产概率,无限时间破产概率及破产概率的局部结果等问题.最后是总结和展望.
Web Markov skeleton process (WMSP for short) is a new class of processes re-cently introduced by Ma's group. Based on the work of Ma's group, it is the first time this thesis introduces the notation of compound web Markov skeleton process (CWMSP for short), then it studies some insurance risk problems under the framework of web Markov skeleton process.
As a hot area of research in risk theory, dependent risk is our focus in this paper. As a new class of processes, it applicable to model the dependence in risk theory, which will be a breakthrough in this area.
Intuitively, a Markov skeleton process is a stochastic process employing a Markov chain as its skeleton. A web Markov skeleton process is a jump process and also a Markov skeleton process in which given information of its skeleton, the time slots between jumps are conditionally independent to each other. The system of a WMSP can be described as follows: where{Xn, n≥0} is a Markov chain, and{Tn, n≥0} is the set of time slots between adjacent jumps.
A compound web Markov skeleton process (CWMSP for short) can be written as where Nt is the associated counting process, which denotes the number of jumps before t, and Yi is the accompaniment of ith jump, which is a r.v.. The system of a CWMSP can be described as follows:
The system described by CWMSPs appear in various natural and social sciences, such as biology, finance, queueing theory, insurance and so on. In particular, CWMSP is a suitable framework for modeling the aggregate amount of claims when the claim sizes and inter-claim times are not independent.
This thesis has five parts:The first part is the introduction and preliminaries, which are the main content of Chapter1. In the second part we introduce WMSP and CWMSP, and give the precise large deviation for CWMSP under some condition. And this part consists of Chapter2and3. The third part is Chapter4, which gives the precise large deviation for the aggregate amount of claims under some specified condition. The main topic of Chapter5,6and7is the discussion of the ruin problem under a given catastrophe risk model, which includes finite-time ruin probability and infinite time ruin probability. The last part is the conclusion and the future work.
引文
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