Asymptotics Theory Of Random Walks And Applications In Risk Theory

It is well known that random walks are one of the most important objects in probability. In the studies of random walks, the supremum and overshoot of random walks are two important objects, which have some important applications in many fields of applied probability, such as queueing theory, risk theory, branching processes and infinite divisible distributions. This paper will consider applications in risk theory. Indeed, if one uses a renewal risk model to describe the insurance risk of an insur-ance company then, the ultimate ruin probability of the insurance company is a tail dsitribution at the initial capital reserve of the supremum of a random walk and the local ruin probability is the local probability of the overshoot of the random walk at an interval. The ultimate and local ruin probabilities measure the risk of an insurance company. Therefore, They are important objects, which the insuers and researchers pay more attention to. Since the overshoot reflects the degree of the deficit of an in-surance company, in a, sence, the local ruin probability has more important theoretical significance and applied value than the ultimate ruin probability.We will investigate the asymptotic theory of random walks and applications in risk theory from the following three aspects.Firstly, from the researching process of random walks, we find that in the studies of the asymptotics of the supremum and overshoot of random walks, one often sup-poses that the related distribution belongs to the convolution equivalent distribution class and the obtained results are often presented in a form of a series of equivalent conditions. This will be perfect. But in the realistic situation, there are some other distributions, which do not belong to the convolution equivalent distribution class. For these distributions, how to estimate the asymptotics of the related quantities of ran-dom walks? Of course, we can not discuss this problem for all these distributions. This paper will give a new distribution class, which can contain the light-tailed convolution equivalent distribution class, and use theγ-transform and localization of distributions to find the relation between this light-tailed dsitribution class and some heavy-tailed local distribution classes. We will investigate the asymptotics of the supremum and overshoot of random walks in this new distribution class. The obtained results can contain the classical result.Secondly, we will discuss the asymptotics of the overshoot of random walks. We mainly consider the case that the related distribution is not the convolution equivalent distribution and give the uniform asyniptotics of the overshoot of random walks. As applications in risk theory, the asymptotic estimates of the local ruin probability in renewal risk model are presented. Meanwwhile, by using the renewal equations, the equivalent conditions of the local asymptotics of the overshoot of random walks are given.Finally, in risk theory there are various risk models to deal with the complicated insurance risk. In most of models, one often supposes that the claim sizes and the claim inter-arrival times arc independent random variables, respectively. But in some realistic situations, they are not independent and they will have some dependence struc-tures. This paper will introduce a new dependence structure. Under this dependence structure, we will consider the asymptotics of the finite-time ruin probability of a risk model with a constant interest rate. The obtained asymptotics are uniform for time in a finite interval.