發(fā)布時(shí)間：2018-06-02 03:10

  本文選題：一維擴(kuò)散過程 + 逸出時(shí)　； 參考：《曲阜師范大學(xué)》2014年碩士論文

【摘要】：近年來,分紅問題在精算數(shù)學(xué)中受到了廣泛的關(guān)注.本文考慮了幾類風(fēng)險(xiǎn)模型的分紅問題.文中討論的風(fēng)險(xiǎn)模型有一維擴(kuò)散風(fēng)險(xiǎn)模型,經(jīng)典風(fēng)險(xiǎn)模型和帶擾動(dòng)的經(jīng)典風(fēng)險(xiǎn)模型;主要涉及的分紅策略有障礙分紅策略,閾值分紅策略和混合分紅策略;研究的主要問題有對(duì)應(yīng)風(fēng)險(xiǎn)模型的逸出時(shí),破產(chǎn)前期望折現(xiàn)分紅,破產(chǎn)前折現(xiàn)分紅的矩母函數(shù)和各階矩,Gerber-Shiu函數(shù)和破產(chǎn)時(shí)的拉普拉斯變換等;用到的工具主要包括逸出時(shí),特殊函數(shù),馬氏過程,泰勒公式和Dynkin公式等.文中的有些問題得到了具體的結(jié)果.根據(jù)文章的具體內(nèi)容本文可分為以下三章: 1)一維擴(kuò)散過程的逸出時(shí)和分紅值函數(shù). 在這一章我們考慮一維時(shí)齊的擴(kuò)散過程在區(qū)間上的逸出時(shí)問題,以及它們?cè)陲L(fēng)險(xiǎn)理論中分紅問題的應(yīng)用.首先,我們利用Dynkin公式推導(dǎo)出了逸出時(shí)的拉普拉斯變換滿足的常微分方程.然后,我們列舉了幾個(gè)在精算學(xué)和金融市場模型中經(jīng)常用到的擴(kuò)散過程,得到了它們的逸出時(shí)的拉普拉斯變換的明確表達(dá)式.最后,我們將逸出時(shí)的拉普拉斯變換與分紅值函數(shù)聯(lián)系起來,利用逸出時(shí)的拉普拉斯變換分別表示出了障礙分紅值函數(shù)和閾值分紅值函數(shù). 2)混合分紅策略下的古典風(fēng)險(xiǎn)模型. 第二章研究了在混合分紅策略下的古典風(fēng)險(xiǎn)模型.我們先詳細(xì)地介紹了古典風(fēng)險(xiǎn)模型、混合分紅策略以及分紅策略問題的研究背景.然后定義了在混合分紅策略下的古典風(fēng)險(xiǎn)模型中的破產(chǎn)前折現(xiàn)分紅函數(shù),破產(chǎn)前期望折現(xiàn)分紅函數(shù),折現(xiàn)分紅函數(shù)的矩母函數(shù)和各階矩, Gerber-Shiu函數(shù)和破產(chǎn)時(shí)的拉普拉斯變換.進(jìn)一步推導(dǎo)出了破產(chǎn)前的期望折現(xiàn)分紅函數(shù)滿足的積分微分方程和邊界條件.我們也得到了破產(chǎn)前折現(xiàn)分紅函數(shù)的矩母函數(shù)及各階矩分別滿足的積分微分方程和邊界條件.最后我們討論了Gerber-Shiu函數(shù)和破產(chǎn)時(shí)的拉普拉斯變換.其中關(guān)于個(gè)體索賠額服從指數(shù)的情況,我們得到了一些具體的結(jié)果. 3)混合分紅策略下的帶擾動(dòng)的古典風(fēng)險(xiǎn)模型. 在第二章的基礎(chǔ)上,本章考慮了在混合分紅策略下的帶擾動(dòng)的古典風(fēng)險(xiǎn)模型.為方便,我們沿用了第二章的部分記號(hào),由于模型的變化破產(chǎn)時(shí)的定義不同,我們首先介紹了帶擾動(dòng)的古典風(fēng)險(xiǎn)模型,指出了由于模型的變化導(dǎo)致的與第二章的記號(hào)的變化.我們利用Dynkin公式推導(dǎo)出了破產(chǎn)前期望折現(xiàn)分紅函數(shù)滿足的積分微分方程和邊界條件.我們?cè)诶�子中得到了�?dāng)索賠服從指數(shù)分布時(shí)破產(chǎn)前預(yù)期折現(xiàn)分紅的具體表達(dá)式.然后分別推導(dǎo)出了破產(chǎn)前折現(xiàn)分紅的矩母函數(shù)和k階矩滿足的積分微分方程和邊界條件.最后,我們討論了著名的Gerber-Shiu函數(shù),具體計(jì)算了當(dāng)個(gè)體索賠額為指數(shù)分布時(shí)破產(chǎn)時(shí)的拉普拉斯變換.另外,本章的一些邊界條件的證明利用了第二章的結(jié)果.
[Abstract]:In recent years, the issue of dividend has received extensive attention in actuarial mathematics. In this paper, the dividend problem of several risk models is considered. The risk models discussed in this paper include one-dimensional diffusion risk model, classical risk model and perturbed classical risk model, including barrier dividend strategy, threshold dividend strategy and mixed dividend strategy. The main problems studied include the expected discounted dividend before bankruptcy, the moment generating function of discounted dividend before bankruptcy, the Gerber-Shiu function of each order and the Laplace transformation of ruin, etc. Special function, Markov process, Taylor formula and Dynkin formula. Some of the problems in this paper have obtained concrete results. According to the specific content of the article, this article can be divided into the following three chapters: 1) the escape time and dividend value function of one dimensional diffusion process. In this chapter we consider the escape time problems of one-dimensional homogeneous diffusion processes on the interval and their application to the dividend problem in risk theory. First, we derive the ordinary differential equation of Laplace transform when escaping by using Dynkin formula. Then we enumerate several diffusion processes often used in actuarial and financial market models and obtain the explicit expressions of Laplace transformation when they escape. Finally, we associate Laplace transform of escape time with dividend value function, and express obstacle dividend value function and threshold dividend value function by using Laplace transformation when escaping. 2) the classical risk model under the mixed dividend strategy. The second chapter studies the classical risk model under the mixed dividend strategy. First, we introduce the classical risk model, the mixed dividend strategy and the research background of dividend strategy in detail. Then we define the discounted dividend function in the classical risk model under the mixed dividend strategy, the expected discounted dividend function before ruin, the moment mother function of the discounted dividend function and each moment, the Gerber-Shiu function and the Laplace transformation at the time of ruin. The integro-differential equations and boundary conditions of the expected discounted dividend function before ruin are derived. We also obtain the moment generating function of the discounted dividend function before ruin and the integro-differential equations and boundary conditions satisfied by each order moment respectively. Finally, we discuss the Gerber-Shiu function and the Laplace transformation in ruin. Among them, we get some concrete results about the individual claim amount from the index. 3) the classical risk model with disturbance under mixed dividend strategy. Based on the second chapter, we consider the classical risk model with disturbance under mixed dividend strategy. For convenience, we use the partial notation of the second chapter. Because the definition of the model is different, we first introduce the classical risk model with disturbance, and point out the change of the symbol caused by the change of the model and the symbol of the second chapter. By using the Dynkin formula, we derive the integro-differential equations and boundary conditions of the expected discounted dividend function before ruin. In this example, we obtain the expression of the expected discount dividend before bankruptcy when the claim service is distributed exponentially. Then the moment generating function of discounted dividend before ruin and the integral differential equation and boundary condition of k-order moment are derived respectively. Finally, we discuss the famous Gerber-Shiu function and calculate the Laplace transformation when the individual claim amount is exponential distribution. In addition, the proof of some boundary conditions in this chapter is based on the results of the second chapter.
【學(xué)位授予單位】：曲阜師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：O211.6;F271;F275

【共引文獻(xiàn)】

相關(guān)期刊論文 前2條

1 王崗;鄭金海;徐龍輝;董文凱;;橢圓形港灣內(nèi)水波共振的解析解[J];工程力學(xué);2014年04期

2 蔡振華;廖新維;尚寶兵;安雷;;致密砂巖氣井的應(yīng)力敏感效應(yīng)滲流模型[J];科技導(dǎo)報(bào);2014年31期

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本文編號(hào)：1967085