Renewal Equations And Asymptotics And Local Asymptotics Of Moments Of Deficit At Ruin

The research about the properties of deficit at ruin in renewal risk models is important, in which many results have been obtained, especially, there are many conclusions about the distribution of deficit; and consequently, some people begin to show interest in the properties of the average deficit in some way, that is the asymptotics of moments. Under the assumption that the distribution of the claim size belongs to the Class S(γ),γ≥0, Cheng et al. (2002) get the asymptotics ofφmoment of deficit by the transform to the ascending ladder height process in the theory of random walk, whereφis a non-negative function satisfying certain conditions. In the second chapter, we will give a new proof of the correponding results with the use of the renewal equation that is a important tool in risk theory and this method is different from the paper . In some conditions, such as when Lundberg exponent exists, the asymptotics of the solutions to renewal equations can be obtained; Recently,Yin et al.(2006) and Wang et al.(2006) gets the asymptotics of the solutions to some renewal equations without Lundberg exponent. In this paper, we investigate the asymptotics ofφmoment of deficit along their route. In the third chapter, the asymptotics of localφmoment will be discussed, in which the local ruin probablity becomes a special case. For the con-viences of studying the above contents, this paper will introduce risk models, renewal equations and some concepts of common distributions in the first chapter.