Large Deviations And Ruin Probabilities For The Risk Model With Dependent Claims

Since the1960’s, heavy-tailed distributions have been widely used in applicationprobability, including branching processes, queueing theory, risk theory and otherfields. On insurance industry, many great risks is caused by large claims, for exam-ple the fire insurance, the storm insurance and the earthquake insurance etc. Andbecause heavy-tailed distributions can describe this characteristics of aggregateclaims. Therefore, it is necessary to study on occurrence regularity of heavy-taileddistributions, and this provided a theory tool for risk evaluation and prediction oninsurance management process. Meanwhile, in the early insurance risk, claim sizesand inter-arrival times were treated as independent, identically distributed randomvariables. However, in our daily life, there may exist dependent relations betweenthem.So, in this paper, we still regard the heavy-tailed distributions as the mainobject and discuss the precise large deviations and asymptotic of the ruin prob-ability. In Chapter1, we introduce some heavy-tailed distributions, the definitionof dependent and the present research situation of heavy-tailed dependent randomvariables. In Chapter2, we establish a customer-arrival-based risk model, in whichclaim size is described as indicator function. Under the assumption that claim sizesrandom variables are negatively dependent, identically distributed and belonged toclass D L, we study the precise large deviations result for loss process both thepartial sums and the random sums. Furthermore, we obtain asymptotic formula forfinite-time ruin probability of the surplus precess. Large deviation probabilities canbe applied in the context of large claim insurance, in particular, reinsurance. We re-mark that results of precise large deviations for random sums are particularly usefulfor evaluation of some risk measures such as conditional tail expectation and value at risk of aggregate claims of a large insurance portfolio. In risk model, the studyof the ruin probability can provide an alarm for decision maker of insurance compa-nies, also is a measure of finance risk, thereby, the study of it has important guidingsignificance to management of insurance companies. In Chapter3, let claim sizes bea sequence of negatively dependent, identically distributed random variables, thereis a measurable function, claim sizes and inter-arrival times obey a dependent struc-ture. Assuming that the claim size distribution obeys class D L, we fist establishweak asymptotic equivalent formula for the finite-time ruin probabilities. Further-more, we obtain a uniform result for a continuous-time renewal risk model with aconstant force of interest. We find the asymptotic behavior of the ruin probabilityis insensitive to the negatively dependent structure among the heavy-tailed claims.