On A Class Of Risk Models With Delayed Claims

Insurance plays an important role in modern society,so that the study of risk theory has got wide attention.With the development of risk theory's research,the risk models evolved from a classical compound poisson model to a variety of complex models.The prob-lems were considered from simply computing probability of ruin and survival to analyzing Gerber-shiu function and many other ruin-related quantities.The more real to describe the actual situation facing the company,the greater the value of the study.In reality,insurance claims are delayed for various reasons.For an example,a natural catastrophe(such as an earthquake)devastates insurers' homes,so that the insurance company need to pay property to the insurers.If the insurers contract the plague brought by the disaster after a,period of time,there would be the medical payments.Therefore,the study of risk models with delayed claims can not only complete the study of risk theory but also promote the development of insurance business.In this paper,we consider a risk model with delayed claims.The main claim causes the delayed claim with probability q(0 ? q ? 1).The time when the delayed claims occurred is determined by the size of the main claim:If the size of the main claim exceeds a limit,the delayed claim will take place at the time of the next main claim;on the contrary,the delayed claim and the main claim would occur simultaneously.In this paper,we will study a generalized Gerber-Shiu function in the risk model with delayed claims.The generalized Gerber-Shiu function was first.proposed in Li and Lu(2016).In the generalized Gerber-Shiu function,we study a default time Tz(u)which is the time when the surplus process first below a certain level z(z ?u)from the level u(u ? 0)instead of the classical ruin time,and there is a new variate N(Tz(u))added to the generalized Gerber-shiu function.N(Tz(u))is the number of the main claims occurred before Tz(u).Then we derive an expression for the generalized Gerber-Shiu function through an operator.For exponential claim sizes,we present the exact representation of the generalized Gerber-Shiu function.