Research On Ruin Probabilities In An Insurer With Stochastic Investment Returns Under Heavy-Tailed Distribution

Major disaster events often lead to insurance companies making large claims,and heavy-tailed distribution can effectively describe the distribution of these large claims.With the increasing wealth of our society and the more frequent financial activities,the investment returns of insurers have an increasingly significant impact on the ruin probabil-ity of insurance risk models.Using finance and insurance theory along with probability theory,combined with stochastic processes and modern asset pricing theory,the thesis builds the risk process models of insurers with heavy-tailed distribution and stochastic investment returns and subsequently studies the ruin probabilities of insurers by using probability theory and its limit theory,stochastic processes,and stochastic analysis.The specific research content of this thesis is summarized as follows:1.The thesis investigates the uniform asymptotic formula of P{sup_0≤t≤TZ（t）>x}and then obtains the uniform asymptotic formula for ruin probabilities,where Z（t）=0^te^-ξ（s-）dη（s）is a stochastic integral driven by Lévy processes,ξ（t）is a stochastic process with càdlàg paths,andη（t）is a Lévy process.Assume that an insurer invests its surplus in a financial market consisting of a bond and two stocks.The thesis utilizes Cox-Ingersoll-Ross（CIR）interest rate to model the bond and employs stochastic volatility models of the Ornstein-Uhlenbeck type and Heston models to model the two stocks,and then establishes a unidimensional insurance risk model with stochastic investment returns and applies the obtained result for P{sup_0≤t≤TZ（t）>x}to the model to obtain its uniform asymptotic formula for ruin probability.2.The thesis studies the asymptotic behaviors of ruin probabilities with respect to the initial capital in a bi-dimensional perturbed insurance risk model with general invest-ment returns.Assume that the investment return is described by a càdlàg process,and two classes of claims and the inter-arrival times follow the Sarmanov dependence struc-ture.When the claim-size distribution has a regularly varying tail,the thesis derives the asymptotic formulas of ruin probabilities with respect to the initial capital over certain time regions.In addition,if the càdlàg process describing investment returns is chosen as the Lévy process,Vasicek interest rate model,CIR interest rate model,or Heston model,the thesis derives the asymptotic estimates for ruin probabilities under the corresponding investment returns.3.The thesis investigates the uniform asymptotic estimates for ruin probabilities in a reinsurance risk model with general investment returns.Assume that the investment re-turn is described by a càdlàg process,and the claim and the inter-arrival times follow the Asimit-Jones dependence structure.When the claim-size distribution has a dominated and extended rapidly varying tail,the thesis obtains the asymptotic upper bound and asymp-totic lower bound,which holds uniformly in certain time regions.When the claim-size distribution has a consistently varying tail,the thesis derives uniform asymptotic formu-las of ruin probabilities with respect to the initial capital over certain time regions.If the càdlàg process describing investment returns is chosen as the Lévy process,Vasicek interest rate model,CIR interest rate model,Heston model,or the stochastic volatility model,the obtained results are also true.And a counterexample is given to show that the Sarmanov copula is outside the range of the Asimit-Jones dependence structure.4.The thesis studies the uniform asymptotic behaviors of ruin probabilities with respect to the initial capital in a d-dimensional insurance risk model with general invest-ment returns.Assume that the investment return is described by a càdlàg process,and the d classes of claims and the inter-arrival times follow a dependence structure.When the joint distribution of claims has a multivariate regularly varying tail,the thesis derives the asymptotic formulas of three ruin probabilities with respect to the initial capital,which holds uniformly for the entire time horizon.And,if the càdlàg process describing invest-ment returns is chosen as the Lévy process,Vasicek interest rate model,CIR interest rate model,Heston model,or the stochastic volatility model,the obtained results are also true.In addition,numerical simulations are conducted as a means of illustrating the obtained results:as the initial capital increases,the ruin probabilities decrease and the ratios of the simulation results of ruin probability to the asymptotic results of ruin probability become closer to 1.When the càdlàg process describing investment returns is described by Va-sicek interest rate models and d=2,numerical simulations are conducted to show the effects of parameters on the asymptotic estimates of ruin probability.