| Actuarial science is a discipline that applies mathematical and sta-tistical methods to analyze,evaluate and manage insurance business.It mainly studies how to use data and models to measure and control insurance risks,en-suring that insurance companies can operate in the long term in a stable man-ner.In actuarial science,ruin probability is an extremely important research topic.Ruin probability refers to the likelihood that a company or organization will be unable to repay its debts within a certain period of time in the future.It is an important indicator for measuring a company’s solvency and finan-cial risk.In recent years,with the development of actuarial science,how to measure the bankruptcy probability of insurance business with heavy tails and dependent structures has become a core issue of concern for insurance com-panies and scholars.This thesis applies risk theory,probability theory,and stochastic processes to study the asymptotic estimation of ruin probabilities for several types of dependent risk models under heavy-tailed claims,build-ing upon existing research.The main contributions and innovations of this work are as follows:Firstly,The asymptotic estimation of the tail behavior of the random weighted sum and maximum with subexponential dependence is investigated.The primary random variable is a real-valued random sequence with different distributions and satisfies the general dependency structure proposed by Ko and Tang[1].The random weight is another nonnegative,unbounded and arbi-trary dependent random sequence,but is independent of the primary random variable sequence.Thus,an asymptotic formula for the random weighted sum and maximum is obtained.Secondly,when the primary random variables sat-isfy the dependent structure proposed by Geluk and Tang[2],the same result is obtained.On this basis,when the random weights are bounded,a Kesten type inequality for random weighted sums is established.Finally,it is applied to the continuous-time risk model with constant interest force and by-claims and obtained the asymptotic formula of finite-time ruin probabilities.Secondly,we analyze a time-dependent risk model with stochastic re-turn.Assume that the price process of stochastic invest is described by a general geometric L′evy process and the claim sizes and inter-arrival times form a sequence of independent and identically distributed random pairs,with each pair following a general dependence structure proposed by Asimit and Badescu[3].When the claim size distribution is subexponential,a precise lo-cally uniform asymptotic expression for the tail probability of the stochas-tic discounted aggregate claims as well as the finite-time ruin probability is achieved.In addition,some numerical simulations are also performed to illus-trate the accuracy of our asymptotic formulas.Thirdly,we consider a continuous-time two-dimensional renewal risk model with subexponential claims.In this model,the claim size vectors and their inter-arrival times form a sequence of independent and identically dis-tributed random vectors following some general dependence structures.We obtain locally uniformly asymptotic estimates of four types of finite time ruin probabilities,which our result not only weaken the conditions of Jiang et al.[4]but also generalize their results.Furthermore,when the distribution of claims has the nonzero Karamata lower index,we get the globally uniform asymp-totic formula of four types of ruin probabilities.Lastly,we also provide the Coupla function that satisfies the dependent conditions.Fourthly,we study a nonstandard continuous-time two-dimensional risk model with a constant force of interest.The two types of claims follow subex-ponential distributions and satisfy a general dependence structure,and each pair of time intervals is arbitrarily dependent.Under some mild conditions,a locally uniform expression for the finite-time ruin probability is obtained.When the two type of claims are consistently varying-tailed,we obtain the asymptotic formula of the infinite-time ruin probability.Furthermore,assume the each pair of interval-time satisfies negative quadrant dependence,the glob-ally uniform estimate of the ruin probability is obtained.Fifthly,we present a continuous-time bidimensional renewal risk mod-el with two classes of claims,in which every kind of business is assumed to pay two types of claims called the first and second ones,respectively.Sup-pose that the first class of claim vectors form a sequence of independent and identically distributed random vectors following a general dependence struc-ture that shares a common renewal counting process,and the second class of claim vectors,independent of the first class of claim vectors,constitute anoth-er sequence of independent and identically distributed random vectors which arrive according to two different renewal counting process.For such a model,when the claims are assumed to be subexponential or belong to the intersection of long-tailed and dominatedly-varying-tailed distributions,some asymptotic formulas on finite-time ruin probabilities are derived.The obtained results substantially extend some existing ones in the literature.Finally,the conclusion and outlook are presented.We briefly summa-rize the work and innovative points of this thesis and look forward to future research content and directions. |
