几类风险模型中的破产问题及最优控制问题研究
发布时间：2018-08-09 15:04
【摘要】：本论文主要利用更新理论,随机控制理论,马氏过程及鞅论等数学工具,研究了几类风险模型的破产问题和最优控制问题.我们研究的风险模型大致可以分为两类,一类是具有随机收入的离散时间的风险模型,另一类是跳跃扩散风险模型(或称带扰动的复合Poisson风险模型).本论文研究内容的结构安排如下. 1.在第二章中,将经典的离散时间的复合二项风险模型中的固定保费收入的情况推广为一个二项过程,并考虑了保险公司的索赔会产生延迟索赔的现象,即一个具有延迟索赔和随机收入的复合二项风险模型.利用风险过程的平稳独立增量性和母函数的方法,得到了该模型在门槛分红策略下的破产之前的期望折现分红总量的一个显示表达式.另外,还通过两个具体的例子,分析了保险公司的初始资产和次索赔被延迟赔付的概率对破产之前的期望折现分红总量的影响. 2.在第三章中,以第二章的模型为基础,将经典的具有延迟索赔的风险模型中的一个主索赔一定会产生一个次索赔的情况推广为一个主索赔以某个概率产生一个次索赔的情况,即一个具有相依索赔和随机收入的复合二项风险模型.研究了该模型在门槛分红策略下的破产之前的期望折现分红总量,得到的结论能够包含第二章中所得的结果,最后还通过两个具体的例子分析了模型中各参数对破产之前的期望折现分红总量的影响. 3.在第四章中,对第三章的模型再进行深入的研究,用一个状态有限的时齐马氏链去刻画每个单位时间段的折现因子(或利率),这样就推广了以往的常利率的情况,即一个具有相依索赔、随机收入和随机利率的复合二项风险模型.通过对该模型在门槛分红策略下的破产之前的期望折现分红总量的研究,得到了它的一个一般表达式,最后在两个具体的例子中给出了它的解析表达式. 4.第五章中,在经典的复合Poisson风险模型的基础上,用一个标准的Brownian运动去刻画影响风险盈余过程的一些随机因素的干扰,然后讨论了保险公司对其股东和投保人均按阈值分红策略进行分红的情况,分别研究了该模型下的破产之前的期望折现分红总量和Gerber-Shin期望折罚函数,得到了它们所满足的积分-微分方程.先把它们所满足的积分-微分方程转化为与之等同的更新方程,再证明相应的更新方程的解的存在唯一性,最后利用更新迭代的办法,分别获得了它们的一种显示表达式. 5.第六章中,探讨了一个跳跃扩散风险模型的最优投资组合与比例再保险问题.采用不变方差弹性模型去刻画风险资产的价格过程,再保险公司按方差保费原理收取保费.针对现有文献中对跳跃扩散风险模型中的扩散项存在的两种不同的解释,同时讨论了以最终财富期望指数效用最大为目标的最优控制问题,分别得到了两种不同解释下的最优策略及其值函数的精确表达式. 6.第七章中,考虑用一个跳跃扩散风险模型去刻画保险公司的盈余过程,金融市场中的风险资产的价格过程由一个几何Levy过程来驱动.另外,对保险公司向再保险公司购买的比例再保险的自留水平加上了一个合理的限制,应用随机控制理论的方法,不仅得到了最优策略及其值函数的精确表达式,还通过具体的数值例子分析了模型中的不同的参数分别对最优策略的影响. 7.第八章中,假设保险公司的盈余过程服从跳跃扩散风险模型,该保险公司除了可以将其资产投资在Black-Scholes金融市场中,还可以通过购买再保险(或接受新业务)来转移一部分风险.研究了该保险公司和市场之间的双人零和博弈问题,应用随机微分博弈的方法,得到了最优最优策略及其值函数的精确表达式.另外,对跳跃扩散风险模型的扩散逼近情况,也求出了最优策略及其值函数的精确表达式.
[Abstract]:In this paper, we mainly use the updating theory, stochastic control theory, Markov process and martingale theory to study the ruin problems and optimal control problems of several types of risk models. The risk models we study can be roughly divided into two types, one is the discrete time risk model with random income, and the other is the jump diffusion risk model. Type (or perturbed composite Poisson risk model). The structure of this paper is as follows.
1. in the second chapter, the fixed premium income in the classical discrete time compound two term risk model is generalized to a two process, and the phenomenon of the delay claim of the insurance company's claim will be considered, that is, a compound two risk model with delayed claim and random income. An expression of the total amount of expected discounted dividends of the model before the threshold dividend strategy is obtained by the method of the increment and the mother function. In addition, two specific examples are given to analyze the shadow of the initial assets of the insurance company and the probability of the sub claim by the delayed claim on the total amount of the expected discounted dividend before bankruptcy. Ringing.
2. in the third chapter, based on the model of the second chapter, the case of a major claim in the classic risk model with delay claim will be extended to a case of a claim for a major claim with a certain probability, that is, a compound two risk model with dependent claims and random income. The result of the expected discounted dividend of the model under the threshold dividend policy can include the results obtained in the second chapters. Finally, the effects of the parameters of the model on the total amount of the discounted dividend before bankruptcy are analyzed by two specific examples.
3. in the fourth chapter, the model of the third chapter is further studied, and the discounted factor (or interest rate) of each unit time period is depicted with a finite time homogeneous Markov chain with a finite state. Thus, the previous ordinary interest rate is generalized, that is, a compound two risk model with dependent claim, random income and random interest rate. In this model, a general expression of the total amount of expected discounted dividends is obtained before the threshold dividend policy, and its analytic expression is given in two specific examples.
In the 4. fifth chapter, on the basis of the classical compound Poisson risk model, a standard Brownian motion is used to describe the interference of some random factors that affect the risk surplus process. Then the insurance company is discussed for the dividend policy of its shareholders and the insured per person according to the threshold dividend strategy, and the bankruptcy of the model is studied respectively. The expected discounted dividends and the Gerber-Shin expectancy penalty function are obtained, and the integral differential equations which they satisfy are obtained. First, the integral differential equation satisfied by them is converted to the equivalent renewal equation, and then the existence and uniqueness of the solutions of the corresponding renewal equations are proved, and the latter is obtained by means of the renewal iteration. A presentation of the expression.
In the 5. sixth chapter, the optimal portfolio and proportional reinsurance problem of a jump diffusion risk model is discussed. The price process of the risk asset is depicted by the invariant variance elastic model. The reinsurance company charges the premium according to the variance premium principle. In the existing literature, there are two kinds of the existence of the diffusion term in the jump diffusion risk model. In the same explanation, the optimal control problem with the maximum utility of the final wealth expectation index is discussed, and the exact expressions of the optimal strategies and their value functions under two different interpretations are obtained respectively.
In the 6. seventh chapter, a jump diffusion risk model is considered to describe the surplus process of insurance companies. The price process of the risk assets in the financial market is driven by a geometric Levy process. In addition, a reasonable limit is added to the insurance company's ratio of reinsurance to the reinsurance company. The method of theory not only obtains the exact expression of the optimal strategy and its value function, but also analyzes the effect of the different parameters in the model to the optimal strategy by a specific numerical example.
In the 7. eighth chapter, it is assumed that the insurance company's earnings process is subject to the jump diffusion risk model. In addition to investing its assets in the Black-Scholes financial market, the insurance company can also transfer a part of the risk by buying Reinsurance (or accepting new business). The problem of the double zero sum game between the insurance company and the market is studied. By using the method of stochastic differential game, the exact expression of the optimal optimal strategy and its value function is obtained. In addition, the exact expression of the optimal strategy and its value function is also obtained for the diffusion approximation of the jump diffusion risk model.
【学位授予单位】：湖南师范大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：F840.3;O232
，
本文编号：2174470
[Abstract]:In this paper, we mainly use the updating theory, stochastic control theory, Markov process and martingale theory to study the ruin problems and optimal control problems of several types of risk models. The risk models we study can be roughly divided into two types, one is the discrete time risk model with random income, and the other is the jump diffusion risk model. Type (or perturbed composite Poisson risk model). The structure of this paper is as follows.
1. in the second chapter, the fixed premium income in the classical discrete time compound two term risk model is generalized to a two process, and the phenomenon of the delay claim of the insurance company's claim will be considered, that is, a compound two risk model with delayed claim and random income. An expression of the total amount of expected discounted dividends of the model before the threshold dividend strategy is obtained by the method of the increment and the mother function. In addition, two specific examples are given to analyze the shadow of the initial assets of the insurance company and the probability of the sub claim by the delayed claim on the total amount of the expected discounted dividend before bankruptcy. Ringing.
2. in the third chapter, based on the model of the second chapter, the case of a major claim in the classic risk model with delay claim will be extended to a case of a claim for a major claim with a certain probability, that is, a compound two risk model with dependent claims and random income. The result of the expected discounted dividend of the model under the threshold dividend policy can include the results obtained in the second chapters. Finally, the effects of the parameters of the model on the total amount of the discounted dividend before bankruptcy are analyzed by two specific examples.
3. in the fourth chapter, the model of the third chapter is further studied, and the discounted factor (or interest rate) of each unit time period is depicted with a finite time homogeneous Markov chain with a finite state. Thus, the previous ordinary interest rate is generalized, that is, a compound two risk model with dependent claim, random income and random interest rate. In this model, a general expression of the total amount of expected discounted dividends is obtained before the threshold dividend policy, and its analytic expression is given in two specific examples.
In the 4. fifth chapter, on the basis of the classical compound Poisson risk model, a standard Brownian motion is used to describe the interference of some random factors that affect the risk surplus process. Then the insurance company is discussed for the dividend policy of its shareholders and the insured per person according to the threshold dividend strategy, and the bankruptcy of the model is studied respectively. The expected discounted dividends and the Gerber-Shin expectancy penalty function are obtained, and the integral differential equations which they satisfy are obtained. First, the integral differential equation satisfied by them is converted to the equivalent renewal equation, and then the existence and uniqueness of the solutions of the corresponding renewal equations are proved, and the latter is obtained by means of the renewal iteration. A presentation of the expression.
In the 5. sixth chapter, the optimal portfolio and proportional reinsurance problem of a jump diffusion risk model is discussed. The price process of the risk asset is depicted by the invariant variance elastic model. The reinsurance company charges the premium according to the variance premium principle. In the existing literature, there are two kinds of the existence of the diffusion term in the jump diffusion risk model. In the same explanation, the optimal control problem with the maximum utility of the final wealth expectation index is discussed, and the exact expressions of the optimal strategies and their value functions under two different interpretations are obtained respectively.
In the 6. seventh chapter, a jump diffusion risk model is considered to describe the surplus process of insurance companies. The price process of the risk assets in the financial market is driven by a geometric Levy process. In addition, a reasonable limit is added to the insurance company's ratio of reinsurance to the reinsurance company. The method of theory not only obtains the exact expression of the optimal strategy and its value function, but also analyzes the effect of the different parameters in the model to the optimal strategy by a specific numerical example.
In the 7. eighth chapter, it is assumed that the insurance company's earnings process is subject to the jump diffusion risk model. In addition to investing its assets in the Black-Scholes financial market, the insurance company can also transfer a part of the risk by buying Reinsurance (or accepting new business). The problem of the double zero sum game between the insurance company and the market is studied. By using the method of stochastic differential game, the exact expression of the optimal optimal strategy and its value function is obtained. In addition, the exact expression of the optimal strategy and its value function is also obtained for the diffusion approximation of the jump diffusion risk model.
【学位授予单位】：湖南师范大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：F840.3;O232
，
本文编号：2174470
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