几类风险模型的破产问题

【作者】 董华；

【导师】 尹传存；

【作者基本信息】 曲阜师范大学 ， 概率论与数理统计， 2003， 硕士

【摘要】 本论文致力于拓展几种不同的风险模型的破产理论。主要研究了phase-type风险模型与相关风险模型。 索赔相关风险模型，是最具现实意义的一类风险模型，到目前为止，这类风险模型得到了比较广泛的研究，相关风险模型包括时间相关，索赔次数相关和索赔量相关。Ambagaspitiya（1998）通过向量的方法解决了一类索赔次数相关的风险模型，推导出了最大损失量的表达式；Cossette and Marceau（2000）考虑了离散时间下相关是如何影响有限时间的破产概率与调整系数的问题；Yuen，K．C．and Wang（2001）研究了相关索赔次数都是Poisson过程的风险模型，这种情况可转化为经典情况来研究。Yuen，K．C．，Guo，J．Y．and Wu，X．Y．（2002）两类索赔的索赔次数是Erlang（2）和Poisson过程的和，而索赔量是相互独立的相关模型，这种情况可转化总索赔量为相互独立的两类索赔的索赔量的和，在此文中得到了最终破产概率的确切表达式与渐近表达式。在第一章中，我们把Yuen，K．C．，Guo，J．Y．and Wu，X．Y．（2002）模型中的Erlang（2）推广为Erlang（n）．我们得到了罚金折现期望满足的积分-微分方程。在本章中，罚金折现期望，破产概率，破产时的赤字，破产前的瞬间盈余的各阶距的渐近表达式，特别地，在n=2时我们还得到了破产前的瞬间盈余，破产时的赤字以及它们联合分布的表达式。主要结果：定理1.2.2罚金折现期望Φ（u）满足如下的积分微分方程其中，λ=λ₁+λ₂+μΦ₁（u）=integral from n=0 to ∞（Φ（u-x）dF₁（x））,Φ₂（u）=integral from n=0 to ∞（Φ（u-x）dF₂（x））。 定理1.3.2若函数Φ^*（α）除了方程（1.3.1）的极点外，在整个复平面上都是解析的，则 DDas。type风险模型是风险理论中应用比较广泛的一类风险模型．Dicksonand Hipp（1998）研究了索赔时间 间隔服从 Erlang间分布的情况，并于 2001年a un了其罚金折现期望满K一积分－微分方g．Yin．C．C．（2002）考虑了索＊时间间隔服从 Erlang（n）分布时的罚金折现期望以及破产前的瞬间盈余和破产时赤字的各阶矩的渐近表达式．Didson and HIPPp000)研究了一类特殊的 SPareAndersen模型－Phas。tx卯p)风险模型．在第＝章中，我ffl进一步研究T一类特殊的Phase上yPem风险模型，即这类风险摸型索赔时间间隔分布的密度函数材满足一微分方程 人。k”c卜 Ah（）＋ k（t）二 0，Vt＞ 0，其中，人＞0，A三4人，V仰三o．易知任意两个指数分布（参数不一定相等）的卷积的密度函数都满足上条件．在本章中，我们给出了这模型的破产概率Q…)满足一股疵的更新方程．进而求出了破产概率的确切表达式．近来有许多文章研究了重尾分布，次指数分布S和a。)（。三一分布类是其中最重要的的两类．在本章的后半部分，我们讨论了当单个索赔量的分布轻尾分布时破产概率的渐近表达式（当初始盈余值。趋于无穷大时）．我们还给出了单个索赔量的分布（刊，PEa叫（。＞0）类和单个索赔量的分布的可积尾分布属于次指数分布类时，破产概率得渐近表达式．主要结果：定理2．2．二 玻产概率中（…满足股疵更新方程： 山（。。）二冲＊丁）（叫＋…u），。ZO．O77）其中， 丁（l。）＝＿［门一、422k（0）22巩，几；尸（。。）十 AZCk（0队；厂（1川， 4，CZ‘’“’”“’“‘““‘“”’“”““’ T）ltt＝1。、1．、Illlt． ‘＼一’包2一”2一’”1”＼一’ ．47C“定理二力．1 设一R万程K2．3）的负根，则 h＊＊R＊＼pj＊）＝EAAI！ZI．．－－－－－－－－－－－－－－－－－－－－－－．《2．4JI U二、” 厂（R其中，L（。）表示（22川式中的分母，川。）如（2．21）式中所示．定理2．一刀 设PIS则 W（…P tim，＝、．（2．481 。、户l（。）AIC—AZCZk（0）一p” 3定理2．4．8 设一。＜0为；，“…）收敛的横坐标，且满足片O“‘，H＆＜I．若PE川叫，则对所有的。＞0均有 刷u，u＋叫 门＋人C以0）叫（AI·一AZCN0）一川·（1一。－‘”‘） tim Al＝＝WWWW．（2．415） 一K P（。）A押·a。（o。＋y）U·（l一片 eU二7（Z）d二）‘其中，丁（O）士。定理2．3．2所示． 在第三章中，我们书 DICkCOll 1lid H…卜（2000）的模型推广到了更一般的亿况－索赔时间间隔分布的密度函数的n阶导数满足一n阶微分方程，得到了破产前的瞬间盈余及破产时的赤字的分布．主要结果：定理3．2．4 当。＞0时，H…、X）二P（T＜。、U（T一）＜叫U仰二叫表示破产前的瞬间盈余的分布，则 1 八．、。、卜1）”－‘C几瓦。…骂。，听，，X)贝。一X)4X＼ ＊h?更多还原

【Abstract】 This dissert at ion is devoted to t he development of ruin theory in two kinds of risk model. One is the common shock model for which the two claim number processes are correlated, that is. the times between claims of both classes relate to Poisson and Krlang (n) processes. Another is a Phase-type risk model in which the times between claims {T1. = 1. 2. ...} with a density function k(t) satisfying a linear differential equal ion.For the common shock risk model, many authors studies various aspects of the common shock model in recent years. Ambagaspitiya (1998) considered a general method of constructing a vector p (p 2) dependent claim numbers from a vector of independent random variables, and derived formulas to compute the correlated aggregate claim distribution for corresponding common shock model with p dependent classes of business. Cossotte and Marceau (2000) used a discrete-time approach to study how the common shodcaffects the finite-time ruin probabilities and the adjustment coefficient. Yuen.K. C., and Wang (2001) studied common shock risk model in which the two claim number processes are both Poisson processes by transforming into classical model. Yuen, Guo and Wu (2002) studied a more general common shock model for which one claim number process is Poisson process while the other is Erlang(2) process. This model is transformed into another risk model for which two claim number processes are independent. In our first chapter, we consider a more general common shock risk model in which one claim number process is Poisson process and the other is Erlang (n) process. In the first section of this chapter, we show that the expectation of the discounted penalty W(u) satisfies an integro-differential equation from which we derive the Laplace transform of W(u). The expectation of the distribution of the time to ruin (T), the surplus prior to ruin (S(T-)) and the deficit at ruin (S(T)). and we also show their distributions. An asymptotic result for W(u) is presented. In the second section, we show that the probability of ruin satisfies an defective renewal equation and an asymptotic expression as the initial u tends to infinity is obtained.In the last section of Chapter 1, we consider a special case that n = 2, and get somedifferent result as Yuen, Guo and Wu (2002). The main results:Theorem 1.2.2 The function (u)-satisfies the integro-differential equationwhere,Theorem 1.3.2 If *( )s analytic on the complex plane except for the roots ofIn second chapter, we consider a risk process in which inter-arrival times have a phase-type(2) distribution, a distribution with a density k(t) satisfying the following second order linear differential equation:The conditions are satisfied for all convolutions of two exponential distributions (with not necessarily equal means). This distribution is a special phase-type distributions. This risk model is a more general than it which is introduced by Dickson and Hipp (2000). They consider some ruin related problems. They consider the compound geometric representation of the infinite time survival probability, as well as the (defective) distributions of the surplus immediately prior to ruin and of the deficit at ruin. But we consider the asymptotic behavior of ruin probability of this model as the initial surplus u tends to infinity, we also show the probability of ruin satisfies a defective renewal equation. The asymptotic exponential and non-exponential behaviors of the ruin probability are examined. The main results:Thoerem 2.2.1 (v) satisfies a defective renewal equation.where.Thoerem 2.4.1 Let -R denote the negative root of equation (2.2.3) Thenwhere, L( ) denots the denominator of (2.2.2), h(u) be defined in (2.2.1). Theorem 2.4.2 Let P1 S, thenTheorem 2.4.3 Let -v < 0 be the left abscissa of convergence of p*( ) and satisfies evz (z) dz < 1. If P e 5(v), then for any z > 0, thenwhere, (z) be the same as Theorem 2.3.2.In the third chapter, we extend the work of Dickson and Hipp(2000), and consi更多还原