Two Kinds Of Expected Discounted Penalty Functions Withphase-type Interclaim Times

The classical risk model is also called compound poisson risk model because its claim system is compound poisson process.For this model,scholars have conducted extensive research from many angles and obtained many valuable results.In recent years,this risk model combined with reality has been generalized:Dassions and Embrechts proposed the concept.of absolute ruin for the first tiIme in 1989.Albercher,Cheung and Thonhauser put forward the concept of stochastic observation.The Erlang distribution and phase-type distribution which are close to actual situa-tion have been considered when it comes to inter-claim times.The charge of premiun differs in different ciremnstance.However,there is only one research on the promotion of one aspect in the present literature.When the premiums are different.on the condition that the company's capital is positive and negative and in the case of stochastic observation,this paper considering the expected discounted penalty functions with phase-type inter-claim times is of great importance.In the first chapter,the research status of the risk model and the construction of this paper are introduced.In the second chapter.the basic concept of phase-type distribution is presented.In the third chapter.on the basis of classical risk model.the expected discounted penalty functions of absolute ruin risk model with two kinds of premium rates and phase-type inter-claim times are discussed.The corresponding integro-differential equations are obtained.The expressions of the laplace t.ransforms of the expected discounted penalty functions with the initial capital above the zero and below the zero are derived by the divided differences.In the fourth chapter.the expected discounted penalty functions with stochastic observation and phase-type inter-claim times are obtained.When the claims are exponential distribution.for the initial capital below the zero,the solutions to the the expected discounted penalty function are discussed according to the solution to the n-dimension linear differential equations with constant coefficient.and an mumerical example is given:The expressions of the laplace transfornis of the expected discounted penalty functions with the initial capital is non-negative are derived by the divided differences.