经典风险模型中带有投资和贷款利率的相关问题

发布时间：2018-04-08 11:14

本文选题：复合泊松风险模型　切入点：Gerber-Shiu函数　出处：《曲阜师范大学》2014年硕士论文

【摘要】：近几十年来,关于带有随机利率的扰动经典风险模型的研究取得较好的成果.Gerber and Yang (2007)研究了带有投资的扰动风险模型的绝对破产问题,并讨论了当索赔服从指数分布的绝对破产概率问题；Yin and Wen (2013)对Paulsen and Gjessing (1997a)中的模型进行了拓展,将随机利率和经典模型中的索赔的取值范围设为(-∞,∞),并且研究了该模型的矩母函数和分红问题. 在这篇论文中,我们仍然假设索赔量与索赔来到时刻是相互独立的.两个相关的布朗运动分别影响着盈余过程和投资收入过程.然而当盈余为负数的时候,保险公司以常数利率贷款从而保持正常运营；当盈余为正时,保险公司按照一定的比例进行两种投资：有风险的和无风险的投资；当盈余达到一个常数界限时,保险公司为了争取到更多的客户,按照某种策略进行分红.近年来,关于投资收入、贷款和分红的问题吸引着人们大量关注. 这篇论文就是在带有随机利率的经典复合泊松模型中,提出了按常值红利界限分红,研究了绝对破产条件下同时带有风险投资和无风险投资的经典风险模型,该模型中的索赔{Xi}是相互独立的随机变量序列,并且是取值在(0,∞)上,得至Gerber-Shiu函数的积分微分方程,由于直接解出带参数的二阶积分微分方程是有困难的,本文只给出σ12a=2,λ=0,ω(x,y)=l时的二阶微分方程：和它们的解分别为本文还分别讨论了该风险模型的障碍分红和阈值分红策略下的期望折扣分红函数的积分微分方程与矩母函数的积分微分方程,在例题中给出密度函数为p(x)=e-x,x∈(0,∞)上的障碍分红方程,且解出当σ12a3=2,δ=b2=0,λ=a+c时分红方程的解的表达式；在阈值分红中,给出σ12a=2,λ=0时的解. 本论文结构如下：第一章绪论是对本文主要结果的简单介绍；第二章对本文所讨论的风险模型进行了推导,论证和介绍；第三章推导出Gerber-Shiu函数的积分微分方程,找到方程满足的边界条件,给出特殊情形下方程的解并找到常数系数的确切表达式；第四和第五章是障碍分红和阈值分红策略的期望折扣分红函数的积分微分方程,同样找到方程的边界条件并得到特殊情况下的一些结果.
[Abstract]:In recent years, about.Gerber and Yang achieved good results of disturbance of the classical risk model with stochastic interest rate (2007) of the perturbed risk model with investment absolute ruin, and discussed when the claim is exponential absolute ruin probability distribution problem; Yin and Wen (2013) Paulsen and Gjessing (1997A) in the model, the stochastic interest rate and the classical model in the range of claims for (- OO, OO), and studied the model of the moment generating function and dividend problem.
In this paper, we will assume the claim amount and the arrival times are independent of each other. Two related Brown movement affects the surplus process and investment income. However, when the surplus is negative, the insurance company with constant interest rate loans to maintain normal operation; when the surplus is positive, the insurance company two kinds of investment in a certain proportion: risk and non risk investment; when the surplus reaches a constant threshold, the insurance company in order to get more customers, dividends in accordance with a certain strategy. In recent years, the investment income, loan and dividend problem attracts people a lot of attention.
This paper is in the classical compound Poisson model with stochastic interest rates, according to the constant dividend barrier dividend, the classical risk model is studied under the condition of absolute ruin with risk investment and risk investment, this model claims {Xi} is independent of the random variable sequence, and the value is in (0 infinity), too, Integro differential equations to Gerber-Shiu function, the direct solution of Two Order Integro differential equations with parameters is difficult, this paper only gives a 12a=2, a =0 (x, y), Omega two order differential equation: =l and their solutions respectively.
This paper discussed the integral differential equations with integral differential equation and moment generating function of the expected discounted dividends function of the risk model with threshold dividend strategy and dividend barrier under the, in the examples given in the density function p (x) =e-x, X (0, OO) barrier equation, and the solution of the sigma Delta 12a3=2, =b2=0, expressions of dividend equation lambda =a+c; the threshold dividend, given a 12a=2, a =0 solution.
This paper is structured as follows: the first chapter is a brief introduction of the main results of this paper; the second chapter discussed the risk model of the derivation, demonstration and introduction; third chapter derived the Integro differential equations of Gerber-Shiu function, find the equation of boundary condition equation solutions under special conditions and find the exact constant expression the coefficient of fourth; and the fifth chapter is the Integro differential equations of the expected discounted dividends function barrier and threshold dividend strategy, also find the boundary conditions of the equations and obtain some results under special circumstances.

【学位授予单位】：曲阜师范大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：F842.3;F832.4;O211.6

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