發(fā)布時(shí)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(xué)》2014年碩士論文

【摘要】：保險(xiǎn)在當(dāng)今社會(huì)的發(fā)展中發(fā)揮著不容忽視的作用,破產(chǎn)理論是對(duì)破產(chǎn)概率及相關(guān)問(wèn)題的研究,而破產(chǎn)概率則是衡量保險(xiǎn)公司穩(wěn)定性的一個(gè)重要指標(biāo),越是能夠刻畫保險(xiǎn)公司所面臨的真實(shí)情況的風(fēng)險(xiǎn)模型,對(duì)保險(xiǎn)公司的價(jià)值就越大。在現(xiàn)實(shí)中,索賠過(guò)程往往并不是平穩(wěn)獨(dú)立增量過(guò)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(xiǎn)公司有必要為這類索賠建立儲(chǔ)備以防范風(fēng)險(xiǎn)。大多數(shù)關(guān)于延遲索賠風(fēng)險(xiǎn)模型研究均假設(shè)主索賠引起的副索賠的種類只有一種,然而在現(xiàn)實(shí)中,受隨機(jī)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(fēng)險(xiǎn)模型。 本文首先研究了一類復(fù)合泊松風(fēng)險(xiǎn)模型,假設(shè)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(guò)對(duì)全概率公式的應(yīng)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(dá)式。假設(shè)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(dá)式。最后通過(guò)數(shù)值算例分析了不同參數(shù)的變化對(duì)生存概率的影響。 隨后本文研究了具有隨機(jī)保費(fèi)的Gerber-Shiu罰金折現(xiàn)函數(shù),同樣假設(shè)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(duì)應(yīng)的主索賠與設(shè)定的門限值大小的對(duì)比。給出了Gerber-Shiu罰金折現(xiàn)函數(shù)的求解過(guò)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(dāng)Gerber-Shiu罰金折現(xiàn)函數(shù)退化為破產(chǎn)概率時(shí)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(fēng)險(xiǎn)模型,每個(gè)主索賠的發(fā)生均會(huì)引起一個(gè)副索賠。第一類風(fēng)險(xiǎn)模型假設(shè)主索賠的計(jì)數(shù)過(guò)程滿足泊松分布且副索賠延遲發(fā)生與否取決于對(duì)應(yīng)的主索賠與設(shè)定的門限值大小的對(duì)比；第二類風(fēng)險(xiǎn)模型則假設(shè)主索賠到達(dá)時(shí)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(dá)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(xué)位授予單位】：華北電力大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：F840;F224

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本文編號(hào)：1954455