The Classical Risk Model Of Discounted Penalty Function In The Case Of Two Random Observation

The contemporary study of ruin theory, the concept of international famous scholar Han U Gerber and Elias S.W.Shiu proposed the expected discounted penalty function at ruin in the last century. In the risk theory, some interest in the important actuarial variables are a special case of the time of ruin in the expected discounted penalty function.These actuarial variables include ruin probability,the Laplace transform of the time of ruin, the surplus immediately before ruin and the joint distribution of the deficit at ruin.As an important risk measure tool,the Gerber-Shiu function has been widely used in the study of the ruin theory.After research,Gerber[17], Gerber and Shiu[18] will promote the classical risk model for the renewal risk model ofthe expo-nential inter-observation time. Albrecher and Stefan Thonhauser[2] Studied the ruin probability of such models.The classical risk theory, mainly deal with the stochastic risk model in the insurance busi-ness, discuss the ruin time and the ultimate ruin probability.The model is divided into continuous time model and the discrete-time model according to time.continuous time models have many research results. Such as Lungberg inequality and Gamer-Lungberg approximate formula. Then Feller, Gerber, Gordon, Willmot etc used the theory and method of random process, achieved many good results.While there is little research on the discrete time model, such as Alberecher and Thonhauser[1,2] study the expected discounted penalty function problem and the dividend problem of compound Poisson risk model with stochastic observation.Dickson and Hipp [8] respectively discusses the observation time following Erlang (2) at the time of ruin and the ruin probability of risk process.This master’s thesis consists of three primary parts:The first chapter is the introduction, which introduces the research status of risk model. Given the symbols and formulas used in this paper, and explained what they express meanings.The second part, considering the compound Poisson risk model for the expected discounted penalty function of the observation interval of uniform distribution. Firstly, Through the full probability formula gives the audit time interval obey uniform distribution of the expected dis-counted penalty function. Secondly,by considering the incremental Laplace transform, given the discounted penalty function satisfied the renewal equation. Lately,for exponential claims,when the claim density function by using the integral differential equation,calculate the expected dis-counted penalty function.In the third part, considering the compound Poisson risk model for the expected discounted penalty function of the observation interval of mixed exponential distribution. Firstly,through the full probability formula gives the audit time interval obey mixed exponential distribution of the expected discounted penalty function. Secondly,by considering the incremental Laplace transform, given the discounted penalty function satisfied the renewal equation. Then for exponential claims,when given the claim density function, by using the integral differential equation,calculate the expected discounted penalty function. Finally, in the case of bankruptcy related data, using the mathmatic software, obtained the numerical calculation and chart dis-play. Comparison the effect of random observation time and compared with the classical contin-uous observation time.