本文選題：MAP風(fēng)險模型　切入點：微分-積分方程　出處：《曲阜師范大學(xué)》2017年碩士論文

【摘要】：自從 Neuts(1979)首次提出了MAP 風(fēng)險模型(Markovian arrival risk process)以來,此類模型一直是保險精算領(lǐng)域研究的熱點模型之一.這是因為此類模型更加貼近現(xiàn)實情況,它可以描述具有多種不同狀態(tài)的保險公司的盈余過程.在MAP風(fēng)險模型中,多種不同狀態(tài)是由一個齊次不可約的連續(xù)時間馬爾科夫鏈所描述.MAP風(fēng)險模型是一類比較廣泛的風(fēng)險模型,其包括經(jīng)典風(fēng)險模型(狀態(tài)空間只含一個狀態(tài).計數(shù)過程為Poisson過程),Phase-type 風(fēng)險模型.馬氏調(diào)制風(fēng)險模型(Markov-Modulated risk process)等.馬氏調(diào)制風(fēng)險模型與MAP風(fēng)險模型區(qū)別在于:馬氏調(diào)制風(fēng)險模型中的狀態(tài)轉(zhuǎn)移與索賠不能同時發(fā)生;而MAP風(fēng)險模型這兩者是可以同時發(fā)生的.近幾十年來,許多學(xué)者討論了各種MAP風(fēng)險模型,如:Lu和Li (2005)討論了馬氏調(diào)制過程的破產(chǎn)問題;Lu和Tsai (2007)考慮了帶干擾的馬氏調(diào)制風(fēng)險模型;Li et al. (2015)討論了 MAP風(fēng)險模型的破產(chǎn)問題.其他的相關(guān)參考文獻可見: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.本論文是對Li et al. (2015)這篇論文的推廣,研究了帶干擾的MAP風(fēng)險模型的破產(chǎn)問題.模型的狀態(tài)由一個齊次不可約的連續(xù)時間的馬爾科夫鏈{J(t),t≥ 0}所決定.假定此馬爾科夫鏈的狀態(tài)空間為{l,2,…,m}.當(dāng)風(fēng)險過程在t時刻處于狀態(tài)i. = 1,2,…,m)時,模型的保費收入ci,索賠額的大小Xi,干擾項的系數(shù)σi均與狀態(tài)i相關(guān).本文研究了帶擾動的MAP風(fēng)險模型的破產(chǎn)問題.本文的內(nèi)容分為三部分:第一章:主要介紹帶擾動的MAP風(fēng)險模型,齊次不可約連續(xù)時間的馬爾科夫鏈的一些相關(guān)理論以及在本文中所用到的差分方法和一些算子;第二章:研究了帶擾動的MAP風(fēng)險模型破產(chǎn)時的Laplace變換φ(u),給出了此函數(shù)所滿足的積分-微分方程.在第二節(jié)中,在馬爾科夫鏈的狀態(tài)空間只含有兩個狀態(tài)這一特殊情況下,借助Laplace逆變換給出了 φ(u)的一個表達式;第三章:研究了直到破產(chǎn)發(fā)生時,索賠次數(shù)的矩母函數(shù),給出了矩母函數(shù)所滿足的積分-微分方程,最后在一種特殊情況下給出了此矩母函數(shù)的一個表達式.
[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，

本文編號：1730462