几类风险模型的破产概率及最优分红问题

发布时间：2018-01-23 15:08

本文关键词： 障碍策略阈值策略对偶模型最优分红策略谱正Levy过程 Gerber-Shiu函数积分-微分方程首出时　出处：《曲阜师范大学》2014年博士论文　论文类型：学位论文

【摘要】：近年来,金融及保险公司的分红问题已引起人们的广泛关注.公司应该如何付给股东红利?一个可能的目标是能使公司破产前的期望折现红利达到最大.它最初由DeFinetti (1957)提出.近年来,很多学者对分红问题进行了研究.在对分红问题的研究中两种分红策略被广泛研究：障碍(barrier)分红策略和阈值(threshold)分红策略.本文主要利用更新理论、随机控制理论、概率论、鞅论等工具对更新风险模型和Levy风险模型研究上述两种策略下的分红或最优分红问题. 根据内容本文可以分为以下六章：在第一章中,我们介绍了几类风险模型和最优分红的基础知识. 在第二章中,我们考虑了公司在破产前的盈余过程是一谱正的Levy过程.假设当公司盈余超过上限时,就以常数α进行分红,即采用阈值分红策略.首先我们得到期望折扣分红总量满足的积分-微分方程,然后我们得到了期望折扣分红总量的具体表达式,从而得到阈值分红策略是在此模型下的最优策略. 在第三章中,我们研究了具有终值的谱正Levy风险模型的最优分红问题.利用谱正Levy过程的波动理论,我们得到障碍分红策略下值函数的精确解,从而证明障碍分红策略是最优策略. 在第四章中,我们考虑了对偶风险模型的障碍分红问题.首先通过更新理论得到期望折扣分红总量满足的积分-微分方程及其边界条件,并给出两种特殊情况下的具体表达式.最后得到了期望折扣分红总量的矩母函数及各阶矩满足的积分-微分表达式. 在第五章中,我们研究了具有有理拉普拉斯变换的跳跃过程的首出时问题.我们首先利用儒歇定理得到其林德贝格方程的解；然后利用鞅论得到带型区域首出时及首出位置联合分布的表达式；最后利用上述结果得到期望折扣分红总量在障碍分红及阈值分红下的具体表达式. 在第六章中,我们研究了具有随机投资回报的扩展Paulsen-Gjessing风险模型.我们得到此模型的Gerber-Shiu函数及期望折扣分红函数满足的积分-微分方程,然后得到在障碍分红及阈值分红两种策略下的期望折扣分红总量的矩母函数及各阶矩的精确表达式.
[Abstract]:In recent years, the issue of dividends of financial and insurance companies has aroused widespread concern. How should companies pay dividends to shareholders? One possible goal is to maximize the expected discounted dividend before a company goes bankrupt. It was originally proposed by DeFinetti 1957. in recent years. Many scholars have studied the issue of dividend. In the study of dividend, two kinds of dividend strategies have been widely studied: barrier (barrier) and threshold threshold). This paper mainly uses renewal theory. Stochastic control theory, probability theory, martingale theory and other tools are used to study the dividend or optimal dividend under these two strategies for renewal risk model and Levy risk model. According to the content of this article can be divided into the following six chapters: In the first chapter, we introduce several kinds of risk models and the basic knowledge of optimal dividend. In the second chapter, we consider that the earnings process before bankruptcy is a positive Levy process. First, we obtain the integro-differential equation of the total amount of expected discount dividends, and then we obtain the concrete expression of the total amount of expected discount dividends. Thus, the threshold dividend policy is the optimal strategy under this model. In the third chapter, we study the optimal dividend of the spectral positive Levy risk model with final value. By using the wave theory of the spectral positive Levy process, we obtain the exact solution of the value function under the barrier dividend strategy. Thus, it is proved that obstacle dividend strategy is the best strategy. In Chapter 4th, we consider the problem of obstacle dividend in dual risk model. Firstly, we obtain the integro-differential equation and boundary conditions of the expected discounted total dividend by means of renewal theory. Finally, the moment generating function of the expected discounted total dividends and the integral-differential expression of each order moment satisfied are obtained. In Chapter 5th, we study the first time problem of the jump process with rational Laplace transform. Firstly, we obtain the solution of the Linderberg equation by using the Ruch theorem. Then by using martingale theory, the expression of the joint distribution of the first outgoing time and the first out position of the zone is obtained. Finally, by using the above results, the concrete expressions of the total amount of expected discount dividends under the barrier dividend and the threshold dividend are obtained. In Chapter 6th. We study the extended Paulsen-Gjessing risk model with stochastic return on investment. We obtain the Gerber-Shiu function of this model and the expected discount dividend function. Integro-differential equations. Then the moment function of the total expected discount dividend and the exact expressions of each order moment are obtained under the two strategies of barrier dividend and threshold dividend.
【学位授予单位】：曲阜师范大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：F840.31;O211.6

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