The Study Of Asymptotics For Ruin Probabilities In Some Discrete-time Risk Models

A large amount of data have shown that the bankruptcies of insurance companies are mainly caused by huge claims(or huge net insurance losses),which coincide with extreme events.In applied probability theory,such claims(or net insurance losses,i.e.,the total claim amount minus the total premium income)are usually heavy-tailed distributed,in other words,their distributions have high peaks and heavy tails.A heavy-tailed risk model is defined according to the claims(or the net insurance losses)with heavy-tailed distributions.As pointed out by Embrechts et al.(1997),in insurance industry,heavy-tailed(in particular,subexponential)distributions are well recognised as standard models for individual claim sizes.Therefore,it is important not only in theoretical studies but also in practice to investigate all kinds of heavytailed risk models for insurance companies.Insurance risk models include the continuous-time one and discrete-time one.Our paper mainly considers some discrete-time risk models,especially those under some certain dependence structures.There are two new trends of the study to introduce dependence structures to the discrete-time risk model.One is that all insurance risks of an insurance company obey a certain dependence structure,while financial risks are arbitrarily dependent,and insurance risks are independent of financial risks;Another trend is that the insurance and financial risks constitute a sequence of independent and identically distributed random vectors,however,some certain dependence may exist between each pair of the insurance and financial risks.This paper aims to investigate the asymptotic relations of ruin probabilities in some discrete-time risk models,under the above two types of the dependence structures.We mainly focus ourselves on some heavy-tailed insurance risks,while,in the last chapter,we supplement the case of light-tailed insurance risks(i.e.,Gamma-like tailed insurance risks).This paper is organized as follows.Chapter 1 introduces the background of our studies,some related notions and notation we often used in this paper,and looks back at some existing results.Chapter 2 considers the risk model with pairwise asymptotically independent insurance net losses.A lot of classical results require all insurance risks belong to some certain heavy-tailed distribution class.The objects in this chapter are some pairwise asymptotically independent insurance net losses.We use the condition of the average distribution of the insurance net losses belonging to some certain class to describe our model,and obtain the asymptotics for the finitetime ruin probability.In addition,we use the Crude Monte Carlo(CMC)simulations to verify the asymptotic formulas we have obtained in this chapter.Chapter 3 derives the asymptotics for the partial sum generated by some pairwise asymp-totically independent or pairwise upper extended negatively dependent random variables,with the applications to risk theory.This chapter obtains the main results under the above two dependence structures.We further apply our theoretical results to the discrete-time risk model with insurance and financial risks.We remark that our condition is imposed on the distribution of the insurance net losses' maximum.Comparing with Chapter 2,we enlarge the scope of the insurance net losses.The above two chapters consider the first type of the dependent risk model,whereas in Chapters 4 and 5,we study the model under the second type of dependence structure.Based on such a dependence structure,we derive the asymptotic formulas for ruin probabilities.Chapter 4 investigates the interplay of the insurance risks and financial risks.Some traditional results require the condition that the tail of the financial risks are dominated by that of the insurance risks.In this chapter,we cancel the dominant relation between these two kinds of risks,and obtain the asymptotics for finite-time and infinite-time ruin probabilities,holding uniformly for the three cases,in which the insurance risk is more heavy-tailed than,less heavy-tailed than,and equally heavy-tailed to the financial risk.Chapter 5 considers the dependent discrete-time risk model with Gamma-like tailed insurance net losses.Chapters 2–4 study the ruin probabilities in some heavy-tailed risk models,while,in this chapter,we supplement the case of light-tailed insurance net losses,under a more general dependence structure than that in Chapter 4.Besides,We still conduct the numerical computation through the CMC method,and verify the asymptotic formulas obtained in our main results.