Research On The Risk Models With Delayed Claims

Risk theory is one of the core research fields of actuarial science and mathematics,furthermore,bankruptcy theory is the most important part of risk theory,because the bankruptcy theory can be used to judge whether an insurance company is operating normally according to quantitative analysis.So the more accurately a model describes the real situation,the more significant it is to guidance.Most studies about risk theory are based on the assumption that the processes increases steadily and independently,but this assumption is too idealized.From the perspective of occurrences of delayed claims,this thesis studies two cases,one where two kinds of by-claims occur and the other where n kinds of by-claims occur.Firstly,we consider the case where there are two kinds of by-claims in the model.In the Erlang(n)risk model,we assume that the main claim may not induce any kind of byclaims or induce one kind of the by-claims,in addition,the main claim and the by-claim may occur simultaneously,or the occurrence of by-claim may be delayed to the next epoch.Using the auxiliary model and the full probability formula,we can get the integrodifferential equation system of the survival probability,then,on the assumption that the claim amounts are subjected to exponential distributions with the same parameters,we can derive the specific expressions of the survival probability.In order to analyze the influence of different parameters on the model,we perform simulations with different data and obtain the following conclusions:(1)When the other conditions remain the same,the survival probability decrease gradually as the rate of premium of unit time decreases;(2)When the probability that the by-claims and the main claim occur simultaneously decreases,that is,when the total amount of claims is reduced,the survival probability will gradually increase;(3)When the other conditions remain the same,the survival probability gradually decrease with the decrease of the initial surplus.Combining the conclusions with the management policies of insurance companies,this article puts forward some practical suggestions.Further,we consider the case where there are n kinds of by-claims in the model,the risk model is described as follows: the main claim will induce a by-claim,but the byclaim and the main claim may occur simultaneously,or the by-claim may be delayed by a time interval.This thesis studies two risk models with delayed claims.In the first risk model,we assume that the arrival time of main claims obeys Erlang(n)distribution,using the auxiliary model and the total probability formula,we obtain the integro-differential equation system of the survival probability.In the second risk model,we assume that the aggregate main claims process be a compound Poisson process,using the auxiliary model,Lagrange interpolation theorem,the Rouché theorem and Laplace transform,we derived the renewal equation satisfied by the Gerber-Shiu discounted penalty function and obtain the expression of Gerber-Shiu discounted penalty function.Finally,in order to analyze the influence of different parameters,we perform simulations with different data and obtain the following conclusions:(1)With the other conditions unchanged,the survival probability decreases when the initial surplus decreases.(2)With the other conditions unchanged,as the expectation of the distribution function of the by-claims becomes larger,the survival probability gradually decreases.(3)With the other conditions unchanged,when the initial surplus is large enough,even there are by-claims of different strengths,their impact is relatively limited.