Risk Problem Of Insurance With Dependent Structure And Heavy-tailed Claims

In modern risk theory,the insurance business of insurance company is not only an important means of modern economic and social risk management,but also an important part of modern financial system and social security system.The investment business of insurance company is to ensure the stable operation of the insurance business.This is destined to insurance companies have to face the risk of high insurance claims and the potential risk of financial market,namely,insurance risk and financial risk.With the increasing complexity of the economic environment,the measurement and management of the risk of insurance company are facing new challenges.It is the core problem of Modern Actuarial Science to measure the risk of insurance company under the structure of stochastic investment income or dependent insurance and financial risk.The ruin probability is an important index to measure the risk of insurance company.After we introduce the research background and current situation about the risk theory,for three types of the insurance risk model under differently heavy-tailed assumption and the dependent structure,this dissertation will establish asymptotic equivalent formulas and inequalities of the ruin probability of the risk integration process on the initial capital.Firstly,we will study a discrete-time risk model with dependent and heavy-tailed claims,dependent insurance risk and financial risk.In which,we use a unilateral linear process to characterize the claims,assume noises of unilateral linear process are heavytailed,and assume the noise and the discount factor caused by the investment(representing insurance risk and financial risk of the insurance company)have Sarmanov dependent structure.When the distribution of noises respectively belongs to the consistent variation class and the intersection of the dominated variation class and the long-tailed class,the asymptotic estimation formulas of finite-time ruin probability and ultimate ruin probability are obtained by the large deviation theory of randomly weighted sums.The two asymptotic formulas are extended to the regular variation class,and two conservative asymptotic upper bounds are obtained.In the end,we give an example of covering all theorems and corollaries,which makes the results more easily understood.Secondly,we will study the ruin probabilities of a discrete-time risk model,in which it is assumed that the net loss and discount factor caused by investment(representing the insurance risk and financial risk of the insurance company)are dependent,and the distribution of their product belongs to the intersection of dominated variation class and long-tailed class.The asymptotic estimation formulas of the finite-time ruin probability and the ultimate ruin probability are obtained by the large deviation theory of randomly weighted sums.The asymptotic estimations are extended to the consistent variation class and the regular variation class.The conditions for the asymptotic results are greatly simplified when the distribution belongs to the consistent variation class.Further,by changing some conditions,two asymptotic estimation formulas of more easily calculated forms are obtained when the distribution belongs to the regular variation class.Thirdly,we will study the ruin probability of a continuous-time risk model,in which it is assumed that the claim and the interval-arrival time are dependent,the product of claim and the discount factor has distribution belonging to the consistent variation class,the investment process is exponential L?evy process.Through the large deviations theory of randomly weighted sums,we obtain the asymptotic estimation formula of the continuous-time risk model.In this model,the update counting process is also introduced as the claim counting process,and the investment is a combination of risk assets and riskless assets.After that,by changing some conditions,an asymptotic estimation formula is obtained when the distribution belongs to the regular variation class.It is noted that this asymptotic formula has a more easily calculated form.Finally,we will summarize the dissertation and point out the next direction of the research.