Ruin Probabilities With Interest Rates In AR(m) Model

In insurance mathematics, also called actuarial mathematics, ruin theory is the main concept of the risk theory. Also it is one of the popular topics of the risk theory. The research of the ruin theory has hundreds of years history up to now. But most research has not considered interest rates for a long time. Up to now, people have more interest on the research of the ruin probability including fix income of investment. Many researchers take more care of the research of the effect of interest on ruin probability.In classical risk model, the surplus at time n is not dependent with when the premium or claims occurring. As we have not considered interest in classical risk model. But interest is the important part in ruin probability of risk model in real life. It is assumed to be unpractical to make interest rates be a constant or independent identical distributed every year. As we know interest rates in different years are often dependent with each other. But it is very peculiar if we only assumed the interest rates in n_th year to be AR(1) model. Because the interest rates few years before often has great influence on the interest rate in the n_th year, but not only the interest rates one year before.We consider two general classical risk model in this paper, the effects of timing of payments and interest on the surplus process can be included.The new idea of this paper is that the interest rates on the ruin probabilities in the models are assumed to be dependent AR(m) model, (m is positive integer) Generalized Lundberg inequalities for the ruin probability in this model are derived by inductive approaches.In order to explain the reasonability of the upper bounds,we will give an illustrative application in the compound binomial risk process.In the last section of the paper,we consider when the loss distributions have heavy tails,we will give asymptotic formulas for the finite time ruin probability by an inductive approach on the recursive equations.It can be assumed that the part of the content of this paper is the extension of the model of cai(2002). The big distinction of the two is that we assume that the interest rates on the ruin probabilities in the models are assumed to be dependent AR(m) model,that is the interest rate of the m years has an impact on the following interest rate of the n_th year. At the same time,we try that to give asymptotic formulas for the finite time ruin probability of the model on the certain conditions.The model of this paper can be considered as the risk model of the risk theory.