Asymptotic Analysis Of Ruin Probabilities For Risk Models With Heavy-Tailed Distributions And Dependence Structure

Heavy-tailed distributions can be used to describe the probabilities of occurrence of ex-tremes and rare events. In general, the distributions of claim sizes caused by major natural dis-asters, such as earthquakes, hurricanes, tsunami, fire disaster and other destructive phenomena, can be assumed to be heavy-tailed in risk theory. The probabilities of occurrence of these events are very small and it is hardly possible to predict, but once they have occurred, the insurance company will bear huge losses. Hence, risk models with heavy-tailed distribution have already attracted broad attention in applied probability in recent years. Recently, a new trend in the study of risk theory is to introduce various dependence structures to risk models with heavy-tailed dis-tribution. Along this trend, this dissertation investigated ruin probabilities for risk models with heavy-tailed distributions and dependence structure. The main contents include the following aspects.Firstly, we consider a risk model with two correlated classes of insurance business and a constant force of interest. We assume that the correlation comes from a common shock and that the claim-size distribution is heavy-tailed. In reality, the common shock can depict the effect of a natural disaster that causes various kinds of insurance claims. Under this setting, we investigate the tail behavior of the sum of the two correlated classes of discounted aggregate claims. We obtain the uniform asymptotic formulas for some subclass of subexponential distributions.Secondly, we consider the probability of random time absolute ruin in the renewal risk model with constant premium rate and constant force of interest. We assume that claim sizes are conditionally independent on some sigma algebra and that the common distribution belongs to a subclass of subexponential distributions. We obtain the concise asymptotic formula for the subclass. Besides, we also obtain the estimate for sums of conditionally independent and heavy-tailed random variables with nonnegative random weights.Thirdly, we consider the renewal risk model, in which there exists some asymptotic depen-dence relation between claim sizes and the inter-arrival times, and claim sizes are subexponen-tial. Under this setting, we investigate the tail behaviour of random time ruin probability as the initial risk reserve x tends to infinity. We obtain the similar asymptotic formula as the previous results. Finally, we consider the large deviations for randomly weighted sums. We assume that main random variables are a sequence of real-valued random variables with respective different heavy-tailed distributions and widely orthant dependent structure, while random weights are a sequence of nonnegative and bounded random variables. Under some mild conditions, we obtain the large deviation inequalities for determinate sums and random sums. The obtained results extend some ones in extant literature.