Asymptotic Properties Of Risk Models With Dependent Structures

Compared with the risk model under independent assumptions,the risk model under the dependent structure often reflects the insurance practice more truly.However,how to reflect the dependence structure of risk model more appropriately and how to study the complex risk model with dependence structure have been the focus of many scholars’ research on insurance theory in recent years.At present,the research on risk models with dependent structures mainly includes three aspects.The first is the risk model with the same distribution(not necessarily independent)of claim size.Or,in a multidimensional environment,components of the claim amount vectors are allowed to be dependent on each other.Secondly,the interarrival times sequence has a certain dependence structures.Finally,there is a certain dependence structure between the claim sizes and the corresponding claim inter-arrival times.In this thesis,we study the risk model from three aspects: the types of dependent structure,multidimensional risk model,and non-stationary arrival process of claims,thus enriching the research on the risk model with dependence structure.Details are as follows.In Chapter 1,we introduce the classical risk model and summarize the research background and present situation of risk models with dependent structures.After giving the preliminary knowledge and relevant symbols in this chapter,the research content and methods involved are briefly described.In Chapter 2,we study a nonstandard renewal risk model with constant force of interest,in which each main claim induces a delayed claim.Assume that the main claim sizes and the inter-arrival times obey a dependence structure,and so do the delayed claim sizes and the arrival times.Supposing that the distributions of the main and delayed claim sizes both have subexponential tails,asymptotic estimate for the finitetime ruin probability of this risk model is obtained,as the initial surplus tends to infinity.In order to better illustrate the asymptotic estimation of the ruin probability,numerical simulation is also carried out in this chapter.In Chapter 3,we consider a multidimensional risk model,in which an insurer simultaneously confronts m(m≥2)types of claims sharing a common non-stationary and non-renewal arrival process.Assuming that the claims arrival process satisfies a large deviation principle(LDP)and the claim-size distributions are heavy-tailed,asymptotic estimates for two common types of ruin probabilities for this multidimensional risk model are obtained.As applications,we give two examples of the nonstationary point process: a Hawkes process and a Cox process with shot noise intensity,and asymptotic ruin probabilities are obtained for these two examples.In Chapter 4,we discuss a non-standard multidimensional risk model,in which the claim sizes form a sequence of independent and identically distributed random vectors with nonnegative components that are allowed to be dependent on each other.By assuming that the univariate marginal distributions of claim vectors have consistently varying tails,we obtain the precise large deviation formulas for the multidimensional risk model with the regression size-dependent structure.In Chapter 5,we study a bidimensional risk model in which the claim-size vectors and claim inter-arrival times are arbitrarily dependent.The insurer simultaneously confronts two types of claims sharing a common non-stationary arrival process,and the claim-sizes forma sequence of independent and identically distributed random vectors with nonnegative components being dependent on each other.Supposing that the univariate marginal distributions of the claim-size vectors have dominatedly varying tails,precise large deviations for the aggregate amount of claims are obtained.In Chapter 6,we consider a risk model with heavy-tailed claims and Brownian perturbation.Assuming that the distribution function of claim-size is subexponential,and the arrival process of claims is a non-stationary process satisfying the principle of large deviation,the asymptotic formula for the ruin probability of this risk model at a random time is obtained.In the last chapter,the research contents of this thesis are summarized,and the research work and main innovation points are given.The shortcomings and future research contents are also pointed out.