Discounted Penalty Function In The Compound Binomial Model

Research on ruin theory has always been playing a pivot role in the study of risk theory since it bears both an insurance practical background and interests of probability theory. It is well known that the classical ruin theory is based on a Sweden actuary - Lundberg's doctorial thesis, which, however, did not meet the rigid mathematical criteria from a modern point of view. Later, Cramer built Lundberg's previous work on the rigid base of stochastic process theory and developed its model and theorems to the one recognized today.Along with the development of research on ruin theory and rapid change of modern economic and financial environment, research methodology and interests keep on updating and expanding. In this paper, I will first introduce the basic ideas of classical Lundberg-Cramer model as well as some related topics that we are interesting nowadays. Then I will show the two classical proofs of these results. At last, I will present the main results of my research including a recursive formula and an explicit expression for the expected discounted penalty function in the compound binomial model when initial surplus equals 0, by which we can calculate the expectation for given initial surplus other than 0.Additionally, since the (?)(u) considers the surplus before ruin, the deficit at ruin, the discount factor, as well as the time of ruin, it is a more generalized and powerful model. In fact, we can verify that many classical results may be viewed as particular cases of this result and they are demonstrated in the notes and corollaries of this paper.