带干扰风险模型的破产问题研究

【作者】 孔祥立；

【导师】 吕玉华；

【作者基本信息】 曲阜师范大学 ， 概率论与数理统计， 2009， 硕士

【摘要】 古典风险模是人们最早提出的风险模型,也是研究的最为透彻的模型.但该模型过于理想化,现实中有许多干扰项,因此研究带干扰的风险模型是有必要的. Gerber-Shiu罚金折现函数自从1998年由Gerber和Shiu引入到风险理论中以来,一直是人们研究风险理论的热点问题.本文研究了带干扰风险模型的Gerber-Shiu罚金函数和有关的联合分布等问题.本文共分五章:第一章介绍了风险模型发展的历史和现状.第二章介绍了部分有关的概率论知识.第三章针对Barrier分红策略下的干扰风险模型研究了Gerber-Shiu罚金折现函数,首先,利用布朗运动的有关知识首先获得了Gerber-Shiu罚金折现函数满足的积分方程其中(?),然后在此基础上利用m_s(u)的有界性和函数H、h、F的连续可微性求得m_s(u)的连续性和二次连续可微性.第四章在第三章的基础上进一步研究了Barrier分红策略下的带有常数利率的干扰风险模型,首先运用布朗运动的知识我们得到了该模型的Gerber-Shiu罚金折现函数的积分方程其中(?).然后利用m_s(u)的有界性和函数H、h、F的连续可微性求得m_s(u)的连续性和二次连续可微性.最后,我们应用It(?)公式求得了Gerber-Shiu罚金折现函数所满足的积分一微分方程第五章研究了一类随机保费的干扰风险模型的破产问题.设保费率为一任意的离散随机变量ξ(ω),取值为k_i,i=1,2,…,p_i=P(ξ(ω)=k_i).则风险模型可表示为其中ξ(ω)=k_i时,U_i(t):=u+k_it-(?)+W(t), t≥0为带干扰的经典风险模型.我们用随机过程和概率论的方法推导破产概率、末离前最大盈余分布、破产时、破产前瞬时盈余与破产时赤字的联合分布等精算量分布的具体表达式.更多还原

【Abstract】 The classical risk model,which was put forward earlier than other models has been studied most thorough. But it’s an ideal model because there are many disturbances in real life. So it’s very necessary to research the risk model perturbed by diffusion. Gerber-Shiu discounted penalty function became a hot topic from 1998,when it was introduced in risk theories. In this paper, Gerber-Shiu discounted penalty function and joint distributions of some actuarial random variables are discussed.The article is divided into 5 sections.Chapter 1 is about the development and current situation of risk model.Chapter 2 introduces some related stochastic process theories.Chapter 3 discusses Gerber-Shiu discounted penalty function in the risk model perturbed by diffusion with Barrier dividend strategy. First, the integral equation of Gerber-Shiu discounted penalty function is obtained from the knowledge of Brownian motionwhere t₀≤（?）,0≤a≤（?）,On this basis, the properties of continuity and twice-continuous differentiability for m_s（u） are also got from the boundness of m_s（u） and continuous differentiability of the functions H, h and F.Chapter 4 is founded on chapter 3. In this part, we make a further research of the risk model perturbed by diffusion with constant interest rate under Barrier dividend strategy. First, we get the integral equation of Gerber-Shiu discounted penalty function from the knowledge of Brownian motion WhereThen the twice-continuous differentiability of Gerber-Shiu discounted penalty functionis discussed by the boundness of m_s（u） and continuous differentiability of the functions H, h and F.At last, we get its integro-differential equation by the use of Ito’s formulaIn chapter 5, ruin problem of risk model perturbed by diffusion with stochastic premium rate is considered. Suppose the premium rate is any discrete random variableξ（ω）, whose values are k_i, i = 1,2,…. Then the expression of the risk model isWhenξ（ω）=k_i,U_i（t）=u+k_it-（?）X_i+W（t） for t≥0 is the classical modelperturbed by diffusion.Then the exact expressions for actuarial diagnostics, such as the ruin probability, the distribution of extreme surplus before ruin, the joint distribution of the surplus immediately before ruin, the deficit at ruin and the ruin time, are concluded through stochastic process and probability theory methods.更多还原