本文選題：分紅策略 + Gerber-Shiu函數　； 參考：《重慶理工大學》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,提出用期望折現罰函數來研究破產時間的Laplace變換、破產前瞬時盈余和破產后赤字的聯合分布,因而該函數也被稱作Gerber-Shiu函數.隨著風險理論的日益發(fā)展,許多學者嘗試利用該函數去探討一些風險模型中的分紅問題,并取得了相當大的理論成果.因此Gerber-Shiu函數成為研究分紅和破產問題的一個風險度量工具,具有重要的理論研究價值.本文將對風險模型中的三種分紅策略進行研究,并給出Gerber-Shiu函數在這幾類風險模型中的實際應用.主要內容如下:(1)考慮帶干擾經典風險模型中的Barrier分紅策略,分別推導出期望折現分紅函數以及Gerber-Shiu函數所滿足的積分表達式,并證明其關于變量u二次連續(xù)可微,然后利用?Ito公式得出它們所滿足的積分-微分方程,最后給出??b,d?u關于其積分-微分方程特殊形式下解的表達式.(2)研究Threshold分紅策略下帶干擾的廣義Erlang(n)分布的更新風險模型,得到破產前折現分紅總額的矩母函數、Gerber-Shiu函數以及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)風險模型的具體實例.(3)在復合二項風險模型中,將服從線性函數分布的保費收入推廣到服從二項分布的模型中,研究此模型下的隨機分紅問題.利用全概率公式和控制收斂定理,推導出0?u?a和u?a時,Gerber-Shiu函數所滿足的遞推公式,并進一步給出這兩種情況下Gerber-Shiu函數的瑕疵更新方程,最后分別計算出更新方程以及最終破產概率解的表達式,這也是本文的主要創(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.

【學位授予單位】：重慶理工大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：F224

【參考文獻】

相關碩士學位論文 前1條

1 崔冶敏;兩類風險模型的Gerber-Shiu折現罰金函數[D];南京農業(yè)大學;2010年

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本文編號：1860493