發(fā)布時間：2018-05-30 07:28

  本文選題：延遲索賠 + 拉普拉斯變換��； 參考：《華北電力大學》2014年碩士論文

【摘要】：保險在當今社會的發(fā)展中發(fā)揮著不容忽視的作用,破產(chǎn)理論是對破產(chǎn)概率及相關問題的研究,而破產(chǎn)概率則是衡量保險公司穩(wěn)定性的一個重要指標,越是能夠刻畫保險公司所面臨的真實情況的風險模型,對保險公司的價值就越大。在現(xiàn)實中,索賠過程往往并不是平穩(wěn)獨立增量過程,索賠有可能延遲發(fā)生,延遲索賠可以看作是IBNR(Incurred But Not Reported)索賠,保險公司有必要為這類索賠建立儲備以防范風險。大多數(shù)關于延遲索賠風險模型研究均假設主索賠引起的副索賠的種類只有一種,然而在現(xiàn)實中,受隨機因素的影響,主索賠X可能引起副索賠Y,也有可能引起副索賠Z,即主索賠引起的副索賠的種類可能不止一種。基于以上考慮,本文研究了幾類具有延遲索賠的風險模型。 本文首先研究了一類復合泊松風險模型,假設副索賠的種類為兩種,主索賠的發(fā)生可能不引起副索賠,也可能引起兩種副索賠中的一種。通過對全概率公式的應用,首先得到一組生存概率滿足的微積分方程,再利用拉普拉斯變換、拉普拉斯終值定理和儒歇定理,最終可以求得生存概率的表達式。假設主索賠和副索賠滿足相同的指數(shù)分布,得到了生存概率的具體表達式。最后通過數(shù)值算例分析了不同參數(shù)的變化對生存概率的影響。 隨后本文研究了具有隨機保費的Gerber-Shiu罰金折現(xiàn)函數(shù),同樣假設副索賠的種類為兩種。主索賠的發(fā)生分別以一定的概率引起兩種副索賠中的一種,副索賠延遲發(fā)生與否取決于對應的主索賠與設定的門限值大小的對比。給出了Gerber-Shiu罰金折現(xiàn)函數(shù)的求解過程并討論了Gerber-Shiu罰金折現(xiàn)函數(shù)所滿足的瑕疵更新方程,最后給出了當Gerber-Shiu罰金折現(xiàn)函數(shù)退化為破產(chǎn)概率時的數(shù)值算例及分析。 本文最后研究了兩類副索賠的種類為n種的風險模型,每個主索賠的發(fā)生均會引起一個副索賠。第一類風險模型假設主索賠的計數(shù)過程滿足泊松分布且副索賠延遲發(fā)生與否取決于對應的主索賠與設定的門限值大小的對比；第二類風險模型則假設主索賠到達時刻滿足Erlang(2)分布,副索賠以一定的概率延遲發(fā)生。分析得到生存概率的表達式,最后給出了數(shù)值算例及分析。
[Abstract]:Insurance plays an important role in the development of today's society. Bankruptcy theory is the study of bankruptcy probability and related problems, and bankruptcy probability is an important index to measure the stability of insurance companies. The more the risk model can describe the real situation faced by the insurance company, the greater the value to the insurance company. In reality, the claim process is often not a stable independent increment process, claims may be delayed, delay claims can be regarded as IBNR(Incurred But Not Reported) claims, it is necessary for insurance companies to establish reserves for such claims to guard against risks. Most studies on the risk model of delayed claims assume that there is only one category of sub-claims arising from the main claim, but in reality, it is affected by random factors. Main claim X may give rise to sub-claim Yor sub-claim Z. that is, there may be more than one type of sub-claim arising from the main claim. Based on the above considerations, this paper studies several risk models with delay claims. In this paper, we first study a kind of compound Poisson risk model. Assuming that there are two kinds of sub-claims, the occurrence of the main claim may not cause the sub-claim or one of the two kinds of sub-claims. Through the application of the total probability formula, a set of calculus equations of survival probability satisfied is obtained first, and then the expression of survival probability can be obtained by using Laplace transformation, Laplace final value theorem and Ruch theorem. Assuming that the main claim and the sub-claim satisfy the same exponential distribution, the concrete expression of survival probability is obtained. Finally, the influence of different parameters on survival probability is analyzed by numerical examples. Then we study the discounted Gerber-Shiu penalty function with random premium, and assume that there are two kinds of sub-claims. The occurrence of the main claim causes one of the two sub-claims with a certain probability. The delay in the occurrence of the sub-claim depends on the comparison between the corresponding main claim and the threshold value set. In this paper, the process of solving the Gerber-Shiu penalty discounting function is given, and the defect renewal equation satisfied by the Gerber-Shiu penalty discount function is discussed. Finally, the numerical example and analysis of the Gerber-Shiu penalty discounting function when it degenerates to the ruin probability is given. At the end of this paper, we study the risk model of two kinds of sub-claims, each main claim will cause a sub-claim. The first kind of risk model assumes that the counting process of the main claim satisfies the Poisson distribution and the delay of the sub-claim depends on the comparison between the corresponding main claim and the set threshold value. The second type of risk model assumes that the arrival time of the main claim satisfies Erlang2) distribution, and the sub-claim occurs with a certain probability delay. The expression of survival probability is obtained. Finally, numerical examples and analysis are given.
【學位授予單位】：華北電力大學
【學位級別】：碩士
【學位授予年份】：2014
【分類號】：F840;F224

【參考文獻】

相關期刊論文 前7條

1 高建偉;王青壯;陳曉婕;;基于交替更新過程的模糊隨機破產(chǎn)模型[J];經(jīng)濟數(shù)學;2012年02期

2 魏廣華;高啟兵;;常利力下雙復合泊松風險模型破產(chǎn)概率的上界[J];南京師大學報(自然科學版);2009年01期

3 包振華;付永毅;劉志鵬;;特殊索賠下離散時間延遲更新過程的期望貼現(xiàn)罰金函數(shù)[J];遼寧師范大學學報(自然科學版);2013年02期

4 高珊;劉再明;;具有常紅利邊界和延遲索賠的一類離散更新風險模型[J];數(shù)學學報;2011年06期

5 蘭德新;張春生;;一類交錯更新風險過程的罰金函數(shù)[J];天津師范大學學報(自然科學版);2006年03期

6 喬克林,李晉枝,何樹紅;復合二項風險模型的破產(chǎn)概率[J];云南大學學報(自然科學版);2002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險模型的分紅問題[J];應用數(shù)學學報;2011年06期

，

本文編號：1954455