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Studies on the Compound Poisson-Geometric Risk Model

Author: ZhangYing
Tutor: YinChuanCun
School: Qufu Normal University
Course: Probability Theory and Mathematical Statistics
Keywords: Compound Poisson-Geometric risk model Gerber-Shiu discounted penalty function Dividend barrier Integro-differential equations Variable premium income
CLC: F840
Type: Master's thesis
Year: 2009
Downloads: 51
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Abstract


In the classical risk model the claim number follows Poisson processes,the mean is equal to the variance.But in fact,the variance of the claim number is more than the mean,the divergence is relatively bigger.Using the famous expected discounted penalty function,this thesis consider the risk model in which the counting process is the compound Poisson-Geometric risk model（In another country this model is called the P（?）lya-Aeppli risk model）.The thesis is dividend into three chapters according to contents:Chapter 1.Preface.Firstly,we review the development and the generalization of the risk theory,some scholars who devoted themselves to the risk theory and their main results are introduced also.Secondly,we present the compound Poisson-Geometric process and some conclusions that will be used in the following chapters,they are all to be ready for the Chapter 2 and Chapter 3.Chapter 2.In this chapter,we consider the compound Poisson-Geometric risk model with constant dividend barrier.By an idea of[13],the risk model is a stationary renewal risk model.We drive an integro-differential equation for the Gerber-Shiu expected disco（?）nted penalty function.We further prove theft mb,e（u） which is the solution to this equation can be expressed by the penalty function mb（u） with a constant dividend barrier under ordinary renewal risk model.Then we obtain the solution to mb（u） which is in the form of an infinite series.Finally,in some special cases with certain exponential claims,we are able to find closed-form expressions for the Gerber-Shiu expected discounted penalty function.Chapter 3.The compound Poisson-Geometric risk model with variable premium income is introduced.We first consider a new compound Poisson-Geometric risk model in which the insurance premium is a discrete random variable.The Laplace transform of the ruin probability is obtained.Second,we deal with the premium income randomized risk model.The insurance policy arrive with Poisson process and the insurance premium is also a random variable. The aggregate claims process is Poisson-Geometric process.Using martingale method,we discuss the probabilityof ruin and the Lundberg upper bounds is obtained.Then we discuss the integral equation for the penalty function associated with the time of ruin.Explicit results are derived when the claims are exponential distribution.

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CLC: > Economic > Fiscal, monetary > Insurance > Insurance Theory
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