本文關(guān)鍵詞： 比例再保險(xiǎn) 多維跳擴(kuò)散 HJB方程 Lagrange對(duì)偶定理 違約　出處：《上海師范大學(xué)》2014年碩士論文　論文類型：學(xué)位論文

【摘要】：在金融機(jī)構(gòu)中,保險(xiǎn)公司發(fā)揮了越來(lái)越重要的作用.保險(xiǎn)公司在幫助投保人規(guī)避風(fēng)險(xiǎn)的同時(shí)也實(shí)現(xiàn)了自身的盈利,增加了金融市場(chǎng)的效率,在國(guó)民經(jīng)濟(jì)建設(shè)及社會(huì)保障中起到了舉足輕重的作用.保險(xiǎn)公司的管理層如何有效地運(yùn)營(yíng)資本,規(guī)避風(fēng)險(xiǎn),保障金融系統(tǒng)的穩(wěn)定運(yùn)行成為一個(gè)重要課題.本文考慮多維跳擴(kuò)散市場(chǎng)下保險(xiǎn)公司的最優(yōu)控制策略問(wèn)題.保險(xiǎn)公司的管理層通過(guò)選擇控制策略,如再保險(xiǎn)比例、投資策略來(lái)實(shí)現(xiàn)公司價(jià)值最大化、風(fēng)險(xiǎn)最小化等. 本文首先在第一章介紹了保險(xiǎn)公司投資跳擴(kuò)散市場(chǎng)和購(gòu)買比例再保險(xiǎn)的最優(yōu)控制策略問(wèn)題的主要研究方法和現(xiàn)狀,第二章介紹了本文模型所用到的基本預(yù)備知識(shí).在第三章中,保險(xiǎn)公司通過(guò)購(gòu)買比例再保險(xiǎn)來(lái)分散一部分風(fēng)險(xiǎn),同時(shí)又將資產(chǎn)投資于資本市場(chǎng),并且風(fēng)險(xiǎn)資產(chǎn)采用多維跳擴(kuò)散模型.目標(biāo)是選擇最優(yōu)的再保險(xiǎn)比例和投資策略,使得在某一固定的時(shí)刻公司價(jià)值達(dá)到最大,并求出相應(yīng)的值函數(shù).本文首先將多維問(wèn)題轉(zhuǎn)化為一維問(wèn)題,然后利用動(dòng)態(tài)規(guī)劃原理求得最優(yōu)問(wèn)題的HJB方程,最終求得最優(yōu)的再保險(xiǎn)比例和投資策略.并通過(guò)實(shí)例進(jìn)行分析. 在第四章中,保險(xiǎn)公司購(gòu)買比例再保險(xiǎn),并將資產(chǎn)投資十多維跳擴(kuò)散資本市場(chǎng),目標(biāo)是在保險(xiǎn)公司期望終值財(cái)富為定值時(shí),選擇最優(yōu)的投資和比例再保險(xiǎn)策略使終期方差最小,也就是風(fēng)險(xiǎn)最小,并且求得最小方差.為解決這個(gè)問(wèn)題,利用LagrangeX時(shí)偶方法,把均值-方差問(wèn)題看成帶等式約束的最優(yōu)控制問(wèn)題,然后引入Lagrange乘子把原問(wèn)題轉(zhuǎn)化為一個(gè)不帶等式約束的最優(yōu)控制問(wèn)題,從而可以利用動(dòng)態(tài)規(guī)劃的方法進(jìn)行求解,最后關(guān)于Lagrange乘子求最優(yōu)便得到原問(wèn)題的解.最后就得到的結(jié)果進(jìn)行相關(guān)分析討論. 在第五章中,保險(xiǎn)公司購(gòu)買比例再保險(xiǎn),并將資產(chǎn)投資于帶有跳擴(kuò)散的資本市場(chǎng),這里考慮再保險(xiǎn)公司存在違約的可能,那么保險(xiǎn)公司在t時(shí)刻的盈余過(guò)程就需要從兩種情況來(lái)考慮,即t時(shí)刻之前違約和t時(shí)刻之后違約,之后用示性函數(shù)將兩種情況寫(xiě)為一個(gè)隨機(jī)微分方程.目標(biāo)是選擇最優(yōu)的再保險(xiǎn)比例和投資策略,使得在某一固定的時(shí)刻公司價(jià)值達(dá)到最大,并求出相應(yīng)的值函數(shù).求解過(guò)程采用第二章中的方法. 在第六章的結(jié)論與展望中,給出了本文還存在的不足和今后要深入研究的方向.
[Abstract]:Insurance companies play a more and more important role in financial institutions. Insurance companies help policy holders avoid risks, but also achieve their own profits, increasing the efficiency of the financial market. It plays an important role in national economic construction and social security. How the management of insurance companies operate capital effectively to avoid risks. To ensure the stable operation of the financial system has become an important issue. This paper considers the optimal control strategy of insurance companies in the multi-dimensional jump diffusion market. The management of insurance companies choose control strategies such as reinsurance ratio. Investment strategy to maximize the value of the company, risk minimization and so on. In the first chapter, this paper introduces the main research methods and current situation of the optimal control strategy of investment jump diffusion market and purchasing proportional reinsurance of insurance companies. The second chapter introduces the basic preparatory knowledge used in this model. In the third chapter, the insurance company distributes part of the risk by purchasing proportional reinsurance, while investing assets in the capital market. And risk assets use multi-dimensional jump diffusion model, the goal is to select the optimal reinsurance ratio and investment strategy, so that the company value at a fixed time to achieve the maximum. In this paper, the multidimensional problem is first transformed into a one-dimensional problem, and then the HJB equation of the optimal problem is obtained by using the dynamic programming principle. Finally, the optimal reinsurance ratio and investment strategy are obtained. In Chapter 4th, insurance companies buy proportional reinsurance, and invest their assets in ten dimensional jumps to diffuse capital markets, with the goal being when the insurance company expects the ultimate value of wealth to be fixed. In order to solve this problem, the optimal investment and proportional reinsurance strategies are chosen to minimize the terminal variance, that is, the minimum risk, and to obtain the minimum variance. In order to solve this problem, the LagrangeX time-even method is used to solve this problem. The mean-variance problem is regarded as an optimal control problem with equality constraints, and then the Lagrange multiplier is introduced to transform the original problem into an optimal control problem without equality constraints. Finally, the solution of the original problem can be obtained by using the method of dynamic programming. Finally, the solution of the original problem can be obtained by using the Lagrange multiplier. Finally, the correlation analysis and discussion of the obtained results are carried out. In Chapter 5th, insurance companies buy proportional reinsurance and invest their assets in capital markets with a jump diffusion, considering the possibility of reinsurance companies defaulting. Then the earnings process of the insurance company in the t moment needs to be considered from two situations, namely, the default before the t moment and the default after the t moment. The objective is to select the optimal reinsurance ratio and investment strategy so as to maximize the value of the company at a fixed time. The corresponding value function is obtained, and the method in the second chapter is used to solve the problem. In the conclusion and prospect of Chapter 6th, the deficiency of this paper and the direction of further research are given.
【學(xué)位授予單位】：上海師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：F224;F840.31

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