Research On Tail Probabilities Of Aggregate Risk Under Compound Dependence Structures

With the deepening of the research on modern risk theory,the research field of aggregate risk tail probabilities is also expanding.Aggregate risk tail probabilities refer to the tail probabilities of risk portfolio in a period of time.It is widely used in the fields of bankruptcy theory and network information security.Today,with the development of national economy,due to the complexity of insurance and financial system,the classical risk model considering only independent claims is not enough to meet the needs of modern life.Therefore,starting from the current insurance practice,it is of practical significance to consider introducing various dependent structures into insurance risk and financial risk,and then study the aggregate risk tail probabilities with compound dependence structures.According to the number of insurance,this paper studies the aggregate risk tail probabilities with compound dependence structures from two aspects.On the one hand,this paper models the aggregate risk with only a single insurance.Due to the rapid development of investment industry in modern society,many examples show that the square or absolute value of financial asset return is not independent and stable.Moreover,the return on investment of financial assets is always uncorrelated with the volatility factor.Therefore,this paper selects the general adapted càdlàg process without independence and stationarity to describe the return on investment.At the same time,it is assumed that claim follows a one-sided linear process with independent identically distributed step size,and its step size and inter-arrival time obey Sarmanov dependence,so as to construct a single insurance aggregation risk model with compound dependence structures.Finally,when step size of the claim obeys the heavy-tailed distribution,the asymptotic expression of the aggregation risk tail probability of a single insurance under a fixed time T is obtained.On the other hand,by introducing the concept of by-claim,this paper models the aggregate risk including multiple insurance.In order to better fit the current financial environment,this paper uses the stochastic exponential Lévy process to describe the return on investment process.At the same time,it is assumed that the main claim follows a onesided linear process with identically distributed but dependent step size,and step size of the main claim and by-claim obey pairwise strong quasi-asymptotic independence,so as to construct a multi insurance aggregation risk model with compound dependence structures.Therefore,when step size of the main claim and by-claim belong to the same heavytailed distribution,this paper deduces and demonstrates the asymptotic expression of the aggregation risk tail probabilities with consistency in the finite time domain or infinite time domain.Finally,in order to verify the effectiveness of the results,this paper also uses Monte Carlo Method for numerical simulation.