A Study On The Finite Time Ruin Related Variables In Risk Models

As a means of transferring risks,insurance is an effective way to reduce the possible risk losses in the future,so it plays an important role in the economic development.However,with the continuous development of social economy and more and more diversified social activities,the future risks are affected by many internal and external factors,and the risks brought by many factors interweave to bring new challenges to the insurance industry.Facing the complex market environment,how to measure and control the risk effectively is an urgent problem.In the quantitative analysis of risk,it is necessary to establish the corresponding mathematical model and select the appropriate risk measurement index.Insurance companies usually carry out model fitting based on their capital flow status,such as insurance premium income,claim expense,insurance premium and commission of reinsurance expense,and profit brought by investment,to select a more appropriate insurance risk model.And through the analysis of the risk model to make a prediction of its risk,the commonly used indicators such as ruin probability and survival probability.According to the calculated probability of ruin,insurance companies can adopt corresponding ways to carry out risk control.When the survival probability is low,they can adopt external capital injection,adjust the level of insurance premium,and participate in reinsurance.However,in many risk models,the exact value of the survival probability can be calculated only when the claim amount follows a specific distribution.For a general distribution,only the integro-differential equation satisfying the survival probability can be found at present.On the basis of summarizing the existing approximate algorithms of ruin probability and survival probability,the numerical calculation method of survival probability is discussed.At the same time,the Gerber-Shiu function is studied in a more generalized MAP(Markovian Arrival Process)risk model.Firstly,an algorithm is proposed to calculate the finite time survival probability of risk models in Markov environment,and it is applied to approximate the finite time survival probability of Markov-modulated risk model and MAP risk model,and the finite timesurvival probability of participating in reinsurance strategy is also considered.Then,the Gerber-Shiu function in the MAP risk model is studied,which includes the time of ruin,the number of claims up to ruin,and the aggregate amount of claims at the time of ruin.The integral differential equation of the infinite time Gerber-Shiu function is solved by means of Laplace transform and scale function.Finally,this paper discusses the Gerber-Shiu function of finite time in depth.On the basis of solving the infinite time problem,the Lagrange implicit function theorem,the Laplace transform and the scale function are organically combined to solve the finite time case.