本文選題：分紅策略 + Gerber-Shiu函數(shù)　； 參考：《重慶理工大學(xué)》2015年碩士論文

【摘要】：著名精算大師Hans Gerber與Elias Shiu于1998年在“On the time value of ruin”這篇文章中,提出用期望折現(xiàn)罰函數(shù)來(lái)研究破產(chǎn)時(shí)間的Laplace變換、破產(chǎn)前瞬時(shí)盈余和破產(chǎn)后赤字的聯(lián)合分布,因而該函數(shù)也被稱(chēng)作Gerber-Shiu函數(shù).隨著風(fēng)險(xiǎn)理論的日益發(fā)展,許多學(xué)者嘗試?yán)迷摵瘮?shù)去探討一些風(fēng)險(xiǎn)模型中的分紅問(wèn)題,并取得了相當(dāng)大的理論成果.因此Gerber-Shiu函數(shù)成為研究分紅和破產(chǎn)問(wèn)題的一個(gè)風(fēng)險(xiǎn)度量工具,具有重要的理論研究?jī)r(jià)值.本文將對(duì)風(fēng)險(xiǎn)模型中的三種分紅策略進(jìn)行研究,并給出Gerber-Shiu函數(shù)在這幾類(lèi)風(fēng)險(xiǎn)模型中的實(shí)際應(yīng)用.主要內(nèi)容如下:(1)考慮帶干擾經(jīng)典風(fēng)險(xiǎn)模型中的Barrier分紅策略,分別推導(dǎo)出期望折現(xiàn)分紅函數(shù)以及Gerber-Shiu函數(shù)所滿(mǎn)足的積分表達(dá)式,并證明其關(guān)于變量u二次連續(xù)可微,然后利用?Ito公式得出它們所滿(mǎn)足的積分-微分方程,最后給出??b,d?u關(guān)于其積分-微分方程特殊形式下解的表達(dá)式.(2)研究Threshold分紅策略下帶干擾的廣義Erlang(n)分布的更新風(fēng)險(xiǎn)模型,得到破產(chǎn)前折現(xiàn)分紅總額的矩母函數(shù)、Gerber-Shiu函數(shù)以及m階矩的積分-微分方程,并討論它們的邊界條件,最后給出Erlang(2)風(fēng)險(xiǎn)模型的具體實(shí)例.(3)在復(fù)合二項(xiàng)風(fēng)險(xiǎn)模型中,將服從線(xiàn)性函數(shù)分布的保費(fèi)收入推廣到服從二項(xiàng)分布的模型中,研究此模型下的隨機(jī)分紅問(wèn)題.利用全概率公式和控制收斂定理,推導(dǎo)出0?u?a和u?a時(shí),Gerber-Shiu函數(shù)所滿(mǎn)足的遞推公式,并進(jìn)一步給出這兩種情況下Gerber-Shiu函數(shù)的瑕疵更新方程,最后分別計(jì)算出更新方程以及最終破產(chǎn)概率解的表達(dá)式,這也是本文的主要?jiǎng)?chuàng)新之處.
[Abstract]:The famous actuarial master Hans Gerber and Elias Shiu, in the article "On the time value of ruin" in 1998, proposed the use of the expected discounted penalty function to study the Laplace transformation of the bankruptcy time, the joint distribution of the instantaneous surplus before bankruptcy and the deficit after bankruptcy, thus the function is also called the Gerber-Shiu function. With the increasing risk theory, the function is also called. Development, many scholars try to use this function to discuss the problem of dividend in some risk models, and have obtained considerable theoretical results. Therefore, Gerber-Shiu function becomes a risk measurement tool to study the problem of dividend and bankruptcy. It has important theoretical research value. This paper will study three kinds of dividend strategies in the risk model. The practical application of Gerber-Shiu functions in these types of risk models is given. The main contents are as follows: (1) considering the Barrier dividend strategy in the classical risk model with interference, the integral expressions of the expected discounted dividend function and the Gerber-Shiu function are derived respectively, and it is proved that the variable U is continuously differentiable with respect to the variable U. Use the Ito formula to derive the integral differential equation they satisfy, and finally give the expression of the solution in the special form of the integral differential equation of? B, D? U. (2) the renewal risk model of the generalized Erlang (n) distribution with interference under the Threshold dividend strategy is studied, and the moment mother function, Gerber-Shiu function and m order of the total amount of the discounted bonus before the production are obtained. The boundary condition of the moment integral differential equation and their boundary conditions are discussed. Finally, a concrete example of the Erlang (2) risk model is given. (3) in the compound two term risk model, the premium income that obeys the linear function distribution is extended to the model that obeys the two distribution, and the random dividend problem under this model is studied. The full probability formula and control convergence are used. The theorem, derives the recurrence formula which the Gerber-Shiu function satisfies when 0? U? A and u? A, and further gives the defect renewal equation of the Gerber-Shiu function under these two cases. Finally, the expression of the renewal equation and the final ruin probability solution are calculated respectively. This is the main innovation of this paper.

【學(xué)位授予單位】：重慶理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：F224

【參考文獻(xiàn)】

相關(guān)碩士學(xué)位論文 前1條

1 崔冶敏;兩類(lèi)風(fēng)險(xiǎn)模型的Gerber-Shiu折現(xiàn)罰金函數(shù)[D];南京農(nóng)業(yè)大學(xué);2010年

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本文編號(hào)：1860493