Ruin Probability Under Dependence Assumption

In this paper,we address some important topics in insurance and finance,which are closely related to extremal value theory. They are ND with heavy-tailed distribution for ruin probability.Most problems in the risk theory are investigated under the assumption that the claim size distribution is heavy-tailed,so we need to know some property about ND.As an important part of the applied probability theory,large deviation principle is extremely useful in quantitatively describing extremal events. The formulation of the classical large deviation principles contributes to Cramer et al. But the random variables they concerned with had the light-tailed distributions(i.e.,the moment functions of the random variables are finite).However, the heavy-tailed distributions are of great importance in the fields of finance and insurance,and many problems of them come down to one of large deviations(the problem of reinsurance). Therefore,the large deviations of partial sums and random sums of heavy-tailed random variables have become rational objects to the applied probability researchers.Since the 1960s, heavy-tailed distributions have been widely used in branching processes, queueing theory, risk theory including insurance and finance and other fields. In the early researches on insurance and finance, the objects were supposed to be independent, identically distributed random variables. However, in the practical applications, there may exist some dependence among these random variables. And they may not be independent. So, in Chapter 3, we still regard the heavy-tailed distributions, this Chapter obtains the Embrechts-Veraverbeke[1] asymptotic formula for the random walk with dependent steps,where the steps constitute a sequence of negative dependence random variables with a common heavy-tailed distribution.