Estimating Gerber-Shiu Function By Fourier-Cosine Series Expansion Under Different Risk Model

An expected discounted penalty function called Gerber-Shiu function was proposed by Gerber,H.U.and Shiu,E.S.W in 1998 firstly,which is of great significance in the study of risk theory.The importance of it is that it's a comprehensive function,many key problems in the ruin theory,such as ruin probability,ruin deficit,and the claim caused ruin,can be studied via this function.In recent years,it has become a standard risk measure function in the ruin theory and has been widely studied.Since many of the probability characteristics in the risk model are usually unknown,studying the Gerber-Shiu function by statistical methods based on the observed information of the risk market has practical significance.In recent years,this project has aroused extensive attention among scholars.This thesis mainly focuses on how to propose a new non-parametric estimate of the Gerber-Shiu function by Fourier-cosine series expansion method based on the discrete observed information of the surplus process.First of all,this thesis briefly introduces the research status and significance of the Gerber-Shiu function at home and abroad in recent years,as well as the related concepts involved in risk theory.Then,on the basis of the discrete observed information of the surplus process,this thesis introduces the theory and concrete procedures of estimating the Gerber-Shiu function by Fourier-cosine series expansion method.Furthermore,in order to illustrate the wide application of the proposed method,we consider three important extension models of the classical risk model: the risk model perturbed by a Brownian motion,the risk model with stochastic income and the risk model with constant and stochastic income.In each model,we first introduce the method and steps of estimating the Gerber-Shiu function in detail;then to further illustrate the good performance of this method,we study several important forms of the Gerber-Shiu function,and give some numerical simulation results under different assumptions of claim sizes' distribution,such as exponential distribution,Erlang distribution and so on.The corresponding analysis is carried out and compared with the performance of other non-parametric methods that have been proposed;Further,the convergence rate of the estimation for each risk model is obtained.Finally,we summarize the research work in this paper.The results show that the non-parametric method of approximating Gerber-Shiu function by Fourier-cosine sequence expansion method is universally applicable in the study of risk theory.As long as we can get the estimation of the Fourier transform of Gerber-Shiu function under a given risk model,we can further estimate this function.In addition,the non-parametric estimation obtained by this method achieves good approximation effect;it can be easily calculated and the convergence rate is fast.