几类风险模型的分红问题研究

发布时间：2018-06-05 17:18

本文选题：分红策略 + 绝对破产概率　；参考：《中南大学》2013年博士论文

【摘要】：近年来分红策略下的风险模型一直备受精算工作者的关注,它已经成为精算数学当前的研究热点之一.本文考虑几种风险模型的分红问题,研究了相应风险模型的绝对破产概率、累积分红折现均值、累积分红折现的矩母生成函数等特征量,得到了一些具体的结果,根据内容本文分为以下几章：第一章简单回顾了风险理论的发展历程以及分红策略下风险模型的研究现状,并且介绍了本文的主要研究内容与结构安排. 第二章简单介绍了本文的一些约定和基础知识. 第三章研究了常利率和阈值门限分红策略下带干扰的复合泊松风险模型的绝对破产问题,得到了累积分红折现的矩母生成函数和n阶原点矩所满足的积分微分方程及边界条件；进一步得到了此模型下Gerber-Shiu折现罚函数所满足的积分微分方程及相应边界条件,相应地将积分微分方程转化为Volterra型积分方程,最后给出了索赔额为指数分布时绝对破产概率的解析表达式. 第四章研究了考虑流动储备金和常数分红界的复合泊松风险模型的绝对破产问题,推导了到绝对破产时刻累积分红折现的矩母生成函数和n阶原点矩所满足的积分微分方程及边界条件;进一步给出了指数索赔下累积分红折现均值的明确表达式,并通过数值模拟实例探讨了模型中相关参数对累积分红折现均值的影响. 第五章研究了带扰动的常利率和常数分红界下的对偶风险模型,讨论了破产概率和到破产前一时刻累积分红折现均值所满足的积分微分方程,并通过求解合流超几何方程给出了收益为指数分布时累积分红折现均值和破产概率的明确表达式. 第六章研究了随机利率下相依索赔的离散风险模型的分红问题,根据模型假设每次主索赔可能引起一次副索赔,而每次副索赔有可能延迟发生,当资产盈余达到红利界值b时,公司给投保者分发一定红利,考虑预期红利的现值时,假设利率服从一有限状态空间的马尔可夫链,得到了破产前累积分红折现均值所满足的差分方程及特殊索赔情形下累积分红折现均值的精确表达式,并结合实例进行了数值模拟. 第七章研究了常数分红界下两离散相依险种风险模型的分红问题.模型假定一个险种的主索赔以一定的概率引起另外一险种的副索赔,且副索赔可能延迟发生,推导了到破产前一时刻为止累积分红折现均值满足的差分方程,并得到了特殊索赔额下累积分红折现均值的具体表达式,最后结合实际例子进行了数值模拟. 第八章对本文进行了简单的总结,并对后续工作进行了展望.
[Abstract]:In recent years, the risk model under dividend strategy has been paid more attention by actuaries, and it has become one of the current research hotspots in actuarial mathematics. In this paper, we consider the dividend problem of several risk models, study the absolute ruin probability of the corresponding risk model, the average value of the cumulative dividend discount, the moment generating function of the cumulative dividend discount, and obtain some concrete results. According to the content of this article is divided into the following chapters: The first chapter briefly reviews the development of risk theory and the research status of risk model under dividend strategy, and introduces the main research content and structure of this paper. The second chapter briefly introduces some conventions and basic knowledge of this paper. In chapter 3, we study the absolute ruin of the complex Poisson risk model with disturbance under the constant interest rate and threshold dividend strategy, and obtain the integral differential equation and boundary conditions satisfied by the moment generating function of the cumulative dividend discount and the n-order origin moment. Furthermore, the integro-differential equation and the corresponding boundary conditions of Gerber-Shiu discount penalty function under this model are obtained, and the integrodifferential equation is transformed into Volterra type integral equation accordingly. Finally, an analytical expression of absolute ruin probability is given when the claim amount is exponential distribution. In chapter 4, we study the absolute ruin of the compound Poisson risk model considering the current reserve and the constant dividend boundary. In this paper, the moment generating function of cumulative dividend discounted at absolute ruin time and the integral differential equation and boundary condition satisfied by n-order origin moment are derived, and the explicit expression of the average value of cumulative dividend discount under exponential claim is given. The effect of relevant parameters on the average value of cumulative dividend is discussed by numerical simulation. In chapter 5, we study the dual risk model under the constant interest rate and constant dividend bound, and discuss the integro-differential equation satisfied by the ruin probability and the average value of the cumulative dividend at the moment before the ruin. By solving the supergeometric equation of confluence, the explicit expressions of the discounted average value and ruin probability of cumulative dividend are given when the income is exponential distribution. In chapter 6, we study the dividend problem of discrete risk model of dependent claims under random interest rate. According to the model, we assume that each main claim may cause one sub-claim, and each sub-claim may be delayed, when the asset surplus reaches the dividend bound b, When the company distributes a certain dividend to the insured, considering the present value of the expected dividend, it assumes that the interest rate is served by a Markov chain in a finite state space. The difference equation satisfied by the discounted average of cumulative dividend before bankruptcy and the exact expression of the discounted average value of cumulative dividend in special claim are obtained, and the numerical simulation is carried out in combination with an example. In chapter 7, we study the dividend problem of two discrete dependent insurance risk models under the constant dividend bound. The model assumes that the main claim of one type of insurance may cause the sub-claim of another insurance with a certain probability, and the sub-claim may be delayed. The difference equation of the discounted average value of accumulated dividend until the moment before bankruptcy is derived. The concrete expression of the discounted average value of accumulated dividend under special claim amount is obtained, and the numerical simulation is carried out in combination with an actual example. Chapter 8 gives a brief summary of this paper and prospects the follow-up work.
【学位授予单位】：中南大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：F840.3;F224;O211.67

【参考文献】