带扰动的MAP风险模型的破产问题

发布时间：2018-04-10 08:32

  本文选题：MAP风险模型　切入点：微分-积分方程　出处：《曲阜师范大学》2017年硕士论文

【摘要】：自从 Neuts(1979)首次提出了MAP 风险模型(Markovian arrival risk process)以来,此类模型一直是保险精算领域研究的热点模型之一.这是因为此类模型更加贴近现实情况,它可以描述具有多种不同状态的保险公司的盈余过程.在MAP风险模型中,多种不同状态是由一个齐次不可约的连续时间马尔科夫链所描述.MAP风险模型是一类比较广泛的风险模型,其包括经典风险模型(状态空间只含一个状态.计数过程为Poisson过程),Phase-type 风险模型.马氏调制风险模型(Markov-Modulated risk process)等.马氏调制风险模型与MAP风险模型区别在于:马氏调制风险模型中的状态转移与索赔不能同时发生;而MAP风险模型这两者是可以同时发生的.近几十年来,许多学者讨论了各种MAP风险模型,如:Lu和Li (2005)讨论了马氏调制过程的破产问题;Lu和Tsai (2007)考虑了带干扰的马氏调制风险模型;Li et al. (2015)讨论了 MAP风险模型的破产问题.其他的相关参考文献可见:Breuer (2002), Badescuet et al. (2005, 2007),Badescu(2008),Ren (2008),Cheung 和 Landriaclt (2009), Cheung 和 Feng (2013),Li 和Ren (2013),Feng 和 Shimizu (2014)等.本论文是对Li et al. (2015)这篇论文的推广,研究了带干扰的MAP风险模型的破产问题.模型的状态由一个齐次不可约的连续时间的马尔科夫链{J(t),t≥ 0}所决定.假定此马尔科夫链的状态空间为{l,2,…,m}.当风险过程在t时刻处于状态i. = 1,2,…,m)时,模型的保费收入ci,索赔额的大小Xi,干扰项的系数σi均与状态i相关.本文研究了带扰动的MAP风险模型的破产问题.本文的内容分为三部分:第一章:主要介绍带扰动的MAP风险模型,齐次不可约连续时间的马尔科夫链的一些相关理论以及在本文中所用到的差分方法和一些算子;第二章:研究了带扰动的MAP风险模型破产时的Laplace变换φ(u),给出了此函数所满足的积分-微分方程.在第二节中,在马尔科夫链的状态空间只含有两个状态这一特殊情况下,借助Laplace逆变换给出了 φ(u)的一个表达式;第三章:研究了直到破产发生时,索赔次数的矩母函数,给出了矩母函数所满足的积分-微分方程,最后在一种特殊情况下给出了此矩母函数的一个表达式.
[Abstract]:MAP arrival risk process has been one of the hot models in the field of actuarial insurance since it was first proposed by Neutsberg in 1979.This is because the model is more realistic and can describe the earnings process of insurance companies with different states.In MAP risk model, many different states are described by a homogeneous irreducible continuous time Markov chain. Map risk model is a kind of more extensive risk model, which includes classical risk model (state space contains only one state).The counting process is Poisson process and Phase-type risk model.Markov modulation risk model, Markov-modulated risk process, et al.The difference between the Markov modulation risk model and the MAP risk model is that the state transition and the claim in the Markov modulation risk model cannot occur at the same time, while the MAP risk model can occur at the same time.In recent decades, many scholars have discussed various MAP risk models, such as the ruin of the Markov modulation process in the case of: Lu and Li (2005) and the ruin problem in the Markov modulation process (Lu and Tsai 2007).The bankruptcy of MAP risk model is discussed.Other relevant references can be found in: Breuer, Badescuet et al.Landriaclt, Cheung and Feng, Cheung and Feng 2013, Li and Ren, et al.In this paper, Li et al.In this paper, we study the ruin of MAP risk model with disturbance.The state of the model is determined by a homogeneous irreducible Markov chain with continuous time {J _ T _ T _ t 鈮，

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