本文選題：分紅策略 + 絕對破產(chǎn)概率　； 參考：《中南大學(xué)》2013年博士論文

【摘要】：近年來分紅策略下的風(fēng)險模型一直備受精算工作者的關(guān)注,它已經(jīng)成為精算數(shù)學(xué)當前的研究熱點之一.本文考慮幾種風(fēng)險模型的分紅問題,研究了相應(yīng)風(fēng)險模型的絕對破產(chǎn)概率、累積分紅折現(xiàn)均值、累積分紅折現(xiàn)的矩母生成函數(shù)等特征量,得到了一些具體的結(jié)果,根據(jù)內(nèi)容本文分為以下幾章： 第一章簡單回顧了風(fēng)險理論的發(fā)展歷程以及分紅策略下風(fēng)險模型的研究現(xiàn)狀,并且介紹了本文的主要研究內(nèi)容與結(jié)構(gòu)安排. 第二章簡單介紹了本文的一些約定和基礎(chǔ)知識. 第三章研究了常利率和閾值門限分紅策略下帶干擾的復(fù)合泊松風(fēng)險模型的絕對破產(chǎn)問題,得到了累積分紅折現(xiàn)的矩母生成函數(shù)和n階原點矩所滿足的積分微分方程及邊界條件；進一步得到了此模型下Gerber-Shiu折現(xiàn)罰函數(shù)所滿足的積分微分方程及相應(yīng)邊界條件,相應(yīng)地將積分微分方程轉(zhuǎn)化為Volterra型積分方程,最后給出了索賠額為指數(shù)分布時絕對破產(chǎn)概率的解析表達式. 第四章研究了考慮流動儲備金和常數(shù)分紅界的復(fù)合泊松風(fēng)險模型的絕對破產(chǎn)問題,推導(dǎo)了到絕對破產(chǎn)時刻累積分紅折現(xiàn)的矩母生成函數(shù)和n階原點矩所滿足的積分微分方程及邊界條件;進一步給出了指數(shù)索賠下累積分紅折現(xiàn)均值的明確表達式,并通過數(shù)值模擬實例探討了模型中相關(guān)參數(shù)對累積分紅折現(xiàn)均值的影響. 第五章研究了帶擾動的常利率和常數(shù)分紅界下的對偶風(fēng)險模型,討論了破產(chǎn)概率和到破產(chǎn)前一時刻累積分紅折現(xiàn)均值所滿足的積分微分方程,并通過求解合流超幾何方程給出了收益為指數(shù)分布時累積分紅折現(xiàn)均值和破產(chǎn)概率的明確表達式. 第六章研究了隨機利率下相依索賠的離散風(fēng)險模型的分紅問題,根據(jù)模型假設(shè)每次主索賠可能引起一次副索賠,而每次副索賠有可能延遲發(fā)生,當資產(chǎn)盈余達到紅利界值b時,公司給投保者分發(fā)一定紅利,考慮預(yù)期紅利的現(xiàn)值時,假設(shè)利率服從一有限狀態(tài)空間的馬爾可夫鏈,得到了破產(chǎn)前累積分紅折現(xiàn)均值所滿足的差分方程及特殊索賠情形下累積分紅折現(xiàn)均值的精確表達式,并結(jié)合實例進行了數(shù)值模擬. 第七章研究了常數(shù)分紅界下兩離散相依險種風(fēng)險模型的分紅問題.模型假定一個險種的主索賠以一定的概率引起另外一險種的副索賠,且副索賠可能延遲發(fā)生,推導(dǎo)了到破產(chǎn)前一時刻為止累積分紅折現(xiàn)均值滿足的差分方程,并得到了特殊索賠額下累積分紅折現(xiàn)均值的具體表達式,最后結(jié)合實際例子進行了數(shù)值模擬. 第八章對本文進行了簡單的總結(jié),并對后續(xù)工作進行了展望.
[Abstract]:In recent years, the risk model under dividend strategy has been paid more attention by actuaries, and it has become one of the current research hotspots in actuarial mathematics. In this paper, we consider the dividend problem of several risk models, study the absolute ruin probability of the corresponding risk model, the average value of the cumulative dividend discount, the moment generating function of the cumulative dividend discount, and obtain some concrete results. According to the content of this article is divided into the following chapters: The first chapter briefly reviews the development of risk theory and the research status of risk model under dividend strategy, and introduces the main research content and structure of this paper. The second chapter briefly introduces some conventions and basic knowledge of this paper. In chapter 3, we study the absolute ruin of the complex Poisson risk model with disturbance under the constant interest rate and threshold dividend strategy, and obtain the integral differential equation and boundary conditions satisfied by the moment generating function of the cumulative dividend discount and the n-order origin moment. Furthermore, the integro-differential equation and the corresponding boundary conditions of Gerber-Shiu discount penalty function under this model are obtained, and the integrodifferential equation is transformed into Volterra type integral equation accordingly. Finally, an analytical expression of absolute ruin probability is given when the claim amount is exponential distribution. In chapter 4, we study the absolute ruin of the compound Poisson risk model considering the current reserve and the constant dividend boundary. In this paper, the moment generating function of cumulative dividend discounted at absolute ruin time and the integral differential equation and boundary condition satisfied by n-order origin moment are derived, and the explicit expression of the average value of cumulative dividend discount under exponential claim is given. The effect of relevant parameters on the average value of cumulative dividend is discussed by numerical simulation. In chapter 5, we study the dual risk model under the constant interest rate and constant dividend bound, and discuss the integro-differential equation satisfied by the ruin probability and the average value of the cumulative dividend at the moment before the ruin. By solving the supergeometric equation of confluence, the explicit expressions of the discounted average value and ruin probability of cumulative dividend are given when the income is exponential distribution. In chapter 6, we study the dividend problem of discrete risk model of dependent claims under random interest rate. According to the model, we assume that each main claim may cause one sub-claim, and each sub-claim may be delayed, when the asset surplus reaches the dividend bound b, When the company distributes a certain dividend to the insured, considering the present value of the expected dividend, it assumes that the interest rate is served by a Markov chain in a finite state space. The difference equation satisfied by the discounted average of cumulative dividend before bankruptcy and the exact expression of the discounted average value of cumulative dividend in special claim are obtained, and the numerical simulation is carried out in combination with an example. In chapter 7, we study the dividend problem of two discrete dependent insurance risk models under the constant dividend bound. The model assumes that the main claim of one type of insurance may cause the sub-claim of another insurance with a certain probability, and the sub-claim may be delayed. The difference equation of the discounted average value of accumulated dividend until the moment before bankruptcy is derived. The concrete expression of the discounted average value of accumulated dividend under special claim amount is obtained, and the numerical simulation is carried out in combination with an actual example. Chapter 8 gives a brief summary of this paper and prospects the follow-up work.
【學(xué)位授予單位】：中南大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2013
【分類號】：F840.3;F224;O211.67

【參考文獻】

相關(guān)期刊論文 前9條

1 周夢;;具有分紅上界的經(jīng)典風(fēng)險模型的性質(zhì)研究[J];復(fù)旦學(xué)報(自然科學(xué)版);2006年05期

2 王云杰,王過京;帶干擾負風(fēng)險和模型的破產(chǎn)概率[J];高校應(yīng)用數(shù)學(xué)學(xué)報A輯(中文版);2004年04期

3 王春偉;尹傳存;;絕對破產(chǎn)下具有貸款利息及常數(shù)分紅界的擾動復(fù)合Poisson風(fēng)險模型[J];數(shù)學(xué)物理學(xué)報;2010年01期

4 孫景云;;門限分紅策略下復(fù)合Poisson風(fēng)險模型的絕對破產(chǎn)[J];山東大學(xué)學(xué)報(理學(xué)版);2010年03期

5 袁海麗;胡亦鈞;;帶利率和常數(shù)紅利邊界的對偶風(fēng)險模型的研究[J];數(shù)學(xué)學(xué)報;2012年01期

6 劉東海;劉再明;彭丹;;離散的相依風(fēng)險模型的破產(chǎn)問題[J];應(yīng)用數(shù)學(xué)學(xué)報;2009年05期

7 彭丹;侯振挺;劉再明;;隨機利率下相依索賠的離散風(fēng)險模型的分紅問題[J];應(yīng)用數(shù)學(xué)學(xué)報;2011年06期

8 彭丹;侯振挺;劉再明;;常利率和門限分紅策略下帶干擾的泊松風(fēng)險模型的絕對破產(chǎn)問題[J];應(yīng)用數(shù)學(xué)學(xué)報;2012年05期

9 丁芳清;姚定俊;;常利息力影響下跳擴散模型的分紅問題(英文)[J];應(yīng)用概率統(tǒng)計;2011年02期

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