幾類風險模型中的破產(chǎn)問題及最優(yōu)控制問題研究
發(fā)布時間：2018-08-09 15:04
【摘要】：本論文主要利用更新理論,隨機控制理論,馬氏過程及鞅論等數(shù)學工具,研究了幾類風險模型的破產(chǎn)問題和最優(yōu)控制問題.我們研究的風險模型大致可以分為兩類,一類是具有隨機收入的離散時間的風險模型,另一類是跳躍擴散風險模型(或稱帶擾動的復(fù)合Poisson風險模型).本論文研究內(nèi)容的結(jié)構(gòu)安排如下. 1.在第二章中,將經(jīng)典的離散時間的復(fù)合二項風險模型中的固定保費收入的情況推廣為一個二項過程,并考慮了保險公司的索賠會產(chǎn)生延遲索賠的現(xiàn)象,即一個具有延遲索賠和隨機收入的復(fù)合二項風險模型.利用風險過程的平穩(wěn)獨立增量性和母函數(shù)的方法,得到了該模型在門檻分紅策略下的破產(chǎn)之前的期望折現(xiàn)分紅總量的一個顯示表達式.另外,還通過兩個具體的例子,分析了保險公司的初始資產(chǎn)和次索賠被延遲賠付的概率對破產(chǎn)之前的期望折現(xiàn)分紅總量的影響. 2.在第三章中,以第二章的模型為基礎(chǔ),將經(jīng)典的具有延遲索賠的風險模型中的一個主索賠一定會產(chǎn)生一個次索賠的情況推廣為一個主索賠以某個概率產(chǎn)生一個次索賠的情況,即一個具有相依索賠和隨機收入的復(fù)合二項風險模型.研究了該模型在門檻分紅策略下的破產(chǎn)之前的期望折現(xiàn)分紅總量,得到的結(jié)論能夠包含第二章中所得的結(jié)果,最后還通過兩個具體的例子分析了模型中各參數(shù)對破產(chǎn)之前的期望折現(xiàn)分紅總量的影響. 3.在第四章中,對第三章的模型再進行深入的研究,用一個狀態(tài)有限的時齊馬氏鏈去刻畫每個單位時間段的折現(xiàn)因子(或利率),這樣就推廣了以往的常利率的情況,即一個具有相依索賠、隨機收入和隨機利率的復(fù)合二項風險模型.通過對該模型在門檻分紅策略下的破產(chǎn)之前的期望折現(xiàn)分紅總量的研究,得到了它的一個一般表達式,最后在兩個具體的例子中給出了它的解析表達式. 4.第五章中,在經(jīng)典的復(fù)合Poisson風險模型的基礎(chǔ)上,用一個標準的Brownian運動去刻畫影響風險盈余過程的一些隨機因素的干擾,然后討論了保險公司對其股東和投保人均按閾值分紅策略進行分紅的情況,分別研究了該模型下的破產(chǎn)之前的期望折現(xiàn)分紅總量和Gerber-Shin期望折罰函數(shù),得到了它們所滿足的積分-微分方程.先把它們所滿足的積分-微分方程轉(zhuǎn)化為與之等同的更新方程,再證明相應(yīng)的更新方程的解的存在唯一性,最后利用更新迭代的辦法,分別獲得了它們的一種顯示表達式. 5.第六章中,探討了一個跳躍擴散風險模型的最優(yōu)投資組合與比例再保險問題.采用不變方差彈性模型去刻畫風險資產(chǎn)的價格過程,再保險公司按方差保費原理收取保費.針對現(xiàn)有文獻中對跳躍擴散風險模型中的擴散項存在的兩種不同的解釋,同時討論了以最終財富期望指數(shù)效用最大為目標的最優(yōu)控制問題,分別得到了兩種不同解釋下的最優(yōu)策略及其值函數(shù)的精確表達式. 6.第七章中,考慮用一個跳躍擴散風險模型去刻畫保險公司的盈余過程,金融市場中的風險資產(chǎn)的價格過程由一個幾何Levy過程來驅(qū)動.另外,對保險公司向再保險公司購買的比例再保險的自留水平加上了一個合理的限制,應(yīng)用隨機控制理論的方法,不僅得到了最優(yōu)策略及其值函數(shù)的精確表達式,還通過具體的數(shù)值例子分析了模型中的不同的參數(shù)分別對最優(yōu)策略的影響. 7.第八章中,假設(shè)保險公司的盈余過程服從跳躍擴散風險模型,該保險公司除了可以將其資產(chǎn)投資在Black-Scholes金融市場中,還可以通過購買再保險(或接受新業(yè)務(wù))來轉(zhuǎn)移一部分風險.研究了該保險公司和市場之間的雙人零和博弈問題,應(yīng)用隨機微分博弈的方法,得到了最優(yōu)最優(yōu)策略及其值函數(shù)的精確表達式.另外,對跳躍擴散風險模型的擴散逼近情況,也求出了最優(yōu)策略及其值函數(shù)的精確表達式.
[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.
【學位授予單位】：湖南師范大學
【學位級別】：博士
【學位授予年份】：2013
【分類號】：F840.3;O232
，
本文編號：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.
【學位授予單位】：湖南師范大學
【學位級別】：博士
【學位授予年份】：2013
【分類號】：F840.3;O232
，
本文編號：2174470
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