The Study For Related Problems Of Several Kinds Of Omega Risk Models With Variable Premium

In the actual operation of insurance companies,the collection of premiums is affected by many factors such as supply and demand,market economy environment and so on,and it has diversity and complexity.Therefore,considering the single premium in the research of risk model has certain limitations.At the same time,in the study of the classical risk model,it is usually assumed that a company goes out of business as soon as ruin occurs,that is,when the surplus is negative for the first time.In recent years,with the development of the world finance,the openness and utilization rate of insurance companies in terms of capital are getting higher and higher,so that the research on insurance risk theory can no longer be confined to the general concept of ruin in the classical risk model.In order to help insurance companies to prevent and control risks effectively and improve the company's competitiveness,this paper overcome the defect of the single premium and the limitation of the general meaning of the ruin concept,and extend the fixed premium income in the classical compound Poisson risk model to the variable premium whose income depend on the current reserve,and distinguish the concept of ruin(negative surplus)and bankruptcy(going out of business),establish the Omega model with variable premium.And on this basis,the related risk problems of the model were studied.The structure of the research content of this paper is as follows.In chapter 1,firstly,we introduce the research background and the research situationof risk model at home and abroad,including the compound Poisson risk model,the perturbed compound Poisson risk model and the Omega risk model.Secondly,it introduces several common actuarial variables,such as the dividend strategy,expected discounted dividend function,Gerber-Shiu expected discounted penalty function and bankruptcy probability.Finally,the main research results and innovations of this paper are summarized.In chapter 2,the Gerber-Shiu expected discounted penalty function in the compound Poisson Omega model is considered in the presence of a three-step premium rate.Firstly,by distinguishing whether or not the first claim occurs before bankruptcy and considering the time and the amount of the first claim if occurs,then using the strong Markov property of the risk process,the integral equations,the integro-differential equations and the boundary conditions for the Gerber-Shiu expected discounted penalty function and bankruptcy probability are derived.Secondly,the Gerber-Shiu expected discounted penalty function and bankruptcy probability are determined.Finally,when the claim size is all exponential variable,a special numerical example be presented to verify the relationship between the Gerber-Shiu expected discounted penalty function or bankruptcy probability and initial capital of the insurance company or critical level that causes premium changes.In chapter 3,the Gerber-Shiu expected discounted penalty function in the perturbed compound Poisson Omega model with a two-step premium rate is studied.Firstly,using thestrong Markov property,the integro-differential equations and boundary conditions satisfied by the Gerber–Shiu expected discounted penalty function and bankruptcy probability are derived.Secondly,for a constant bankruptcy rate function,the renewal equations satisfied by the Gerber-Shiu expected discounted penalty function are obtained,and by iteration method,the closed-form solutions of the function are also given.Further,the explicit solutions of the Gerber–Shiu expected discounted penalty function are obtained when the claim size is subject to exponential distribution.Finally,using quantitative methods of mathematics and computer technology,a numerical example is discussed to analyze the influence of some parameters in the model on the Gerber-Shiu expected discounted penalty function and bankruptcy probability.In chapter 4,the dividend problems in the the perturbed compound Poisson Omega model with a two-step premium rate is studied.Firstly,using the strong Markov property,the integro-differential equations and boundary conditions satisfied by the expected discounted dividend payments function and the Gerber–Shiu expected discounted penalty function are derived.Secondly,for a constant bankruptcy rate function,the renewal equations satisfied by the expected discounted dividend payments function and the Gerber-Shiu expected discounted penalty function are obtained,respectively,and by iteration method,their closed-form solutions are also given.Furthermore,the explicit solutions of the two kinds of functions are obtained when the claim size is subject to exponential distribution.Finally,a numerical example is presented to analyze the influence of some parameters in the model on the expected discounted dividend payments function and the Gerber-Shiu expected discounted penalty function and bankruptcy probability.