Some Generalized Compound Risk Models And Ruin Probability

Risk means unknown consequences and losses.For insurance companies,the level of risk management affects their solvency,so it is very important to strengthen the understanding and management of risk to improve operating efficiency.Risk theory is a theory that describes the risk process by constructing mathematical models.In order to describe the risk process more closely to the reality,We studies the following three parts on the basis of relevant achievements in this thesis:Firstly,we model a risk process containing n risk types,including investment factors and random disturbance factors,and assume that both the premium process and the claim process are Poisson-Geometric processes.The basic theory of probability and stochastic process is used to derive the equations satisfying the regulation coefficient and obtain the expression of ruin probability.Assuming that premium and claim process both follow exponential distribution,we design an example to analyze the effects of premium arrival intensity and claim arrival intensity,initial reserve and investment,average premium and average claim on ruin probability.The results show that the initial reserve,total investment,premium arrival intensity and average premium all change inversely with the bankruptcy probability,while the claim arrival rate and average claim amount all change positively with the bankruptcy probability.Secondly,we study the delayed claim model in which the premium arrives as a random process.A new process is constructed according to martingale central limit theorem.We obtain the minimum ruin probability and the optimal investment strategy according to the optimal control theory and HJB equation.Some examples are designed to analyze the relationship between each factor and the minimum probability of bankruptcy and the impact of stock market changes on investment portfolio.The results show that the arrival intensity of claims has a positive relationship with the minimum ruin probability,while the arrival intensity of initial reserve and premium has a negative relationship with the minimum ruin probability.When the stock volatility becomes larger,the insurance company will reduce the stock investment,when the stock yield becomes higher,the insurance company will invest more in the stock;We compare the change of bankruptcy probability in the two cases where premiums arrive at a constant rate and premiums arrive at a random process.It shows that in both cases the bankruptcy probability decreases when average premiums increase,and the bankruptcy probability is relatively higher when premiums arrive at a random process.This indicates that maintaining customer relationship to ensure stable income can reduce operational risks,and it is suggested that insurance companies strive to increase the renewal rate.Thirdly,the delayed claim model of two insurance types is studied,in which the premium arrival process is a random process.This part is a further extension of the model in the second part.It is proved that the surplus process can be approximated to a new stochastic process by using martingale central limit theorem,and the minimum ruin probability and the optimal investment strategy are obtained by using the optimal control theory and HJB equation.An example is designed to analyze the relationship between the claim arrival intensity and the minimum probability of bankruptcy.The results show that the claim intensity of the two insurance types changes in the same direction with the probability of bankruptcy,showing the rule that the higher the claim intensity is,the higher the possibility of bankruptcy is.