The Study On Some Problems Of Ruin Probabilities In Risk Models With Dependent Structures

The core content of risk theory is the quantitative analysis and prediction of risks.Ruin theory,as the major component of risk theory,studies ruin probabilities of risk models and plays an important role in the field of finance and insurance.In recent years,frequent natural disasters such as earthquakes,floods and volcanic eruptions often cause huge claims for such insurance,which are very likely to lead to the bankruptcy of insurance companies.Therefore,the scholars and experts in the insurance and actuarial field focus on the case of large claim sizes.In the classical risk model,it is assumed that there are three independent conditions that the claim sizes are independent,and the claim sizes are independent of the inter-arrival times,and the inter-arrival times are independent.This is only for the convenience of mathematics,but is not common in reality.Due to the nonindependence of many risks,there are correlations among the claim sizes or between the claim sizes and the inter-arrival times.For example,during the process of earthquake,some large-scale natural disasters such as tsunami and landslides are triggered by the accumulation of multiple vibrations at a certain time,which lead to the existence of dependence among the claim sizes.Therefore,risk models with dependent structures are more valuable in study.With the development of the national economy,bank risk-free interest rates,government policy changes,people's consumption levels,people's saving level and so on all affect the premium income of insurance companies and their premiums reinvestment.The influences of these factors on the risk model are mostly reflected in the discount factor,and it is indispensable for the study of the discount factor.However,due to the complexity and diversity of the discount factors,the studies are more tedious,so the general form is considered.This paper is based on the general forms of discount factors,and systematically studies the asymptotic estimations of the ruin probabilities of the insurance risk models.For better studying the influences of different factors in the models and the validity of the asymptotic estimations of ruin probabilities,the numerical simulations of ruin probabilities under different discount factors and the differences between the ruin probabilities and the simulated values are achieved.The main results obtained are as follows.First,we get the study results in the theoretical derivation section.Under the discrete risk model with dependent claims,the claim sizes are assumed to be a one-sided linear process,and the premiums are assumed to be a random process.Consider the discount factor as a general function of the time and constant interest rate,we get the asymptotics of the ruin probability under the condition that the claim sizes have heavy-tailed.Then,Consider the case that the continuous risk model with dependent claims,the claim sizes are assumed to be a one-sided linear process,and there is a dependent structure between the step size and inter-arrival time,and the interest rate is constant.The discount factor is assumed to be a random process,and the asymptotic estimation of its ruin probability is obtained under the condition that the claim sizes follow heavy-tailed distributions.Second,we get the results of numerical simulations of specific practical examples.Based on theoretical research results,numerical simulations are conducted on the respective risk models.Numerical simulations of discrete-time risk models with different constant interest rates and variable interest rates are analyzed and compared,and numerical simulations of continuous-time risk models with different random interest rates are analyzed and compared.