The Estimation Of The Upper And Lower Bounds For Ruin Probability And Deficit Distribution In The Discrete Time Renewal Model

As is known to all, risk theory is an important branch of the modern applied mathematics. Using the knowledge and methods in probability theory, mathematical statistics and stochastic processes, we can establish corresponding mathematical model according to the actual problems in insurance company. Ruin theory is the most important part in risk theory. In recent years, some important ruin quantities such as ruin probability and deficit distribution have been hot issues in the risk theory. However, it is difficult to get explicit expressions for those ruin quantities; an effective method is to give their corresponding estimation of the upper and lower bounds.In this paper we mainly study the estimation of the upper and lower bounds for the ruin probability and the deficit distribution in the ordinary discrete time renewal model. In the first chapter, some relevant research background, research trends and current research of scholars both at home and abroad are reviewed. The second chapter mainly studies ruin probability in the ordinary discrete time renewal model with initial capital u . More precisely, the first section introduces some knowledge of ruin theory, principle and the establishment of ordinary discrete time renewal risk model; in the second section we give some preliminary lemmas by using the defective renewal equations satisfied by ruin probability and deficit distribution; in the third section we give the lower estimation by using lemmas established before; the fourth section shows the corresponding upper bounds of ruin probability. The third chapter investigates the deficit distribution in the ordinary discrete time renewal model with initial capital u . In the first section we obtain double-edge bounds for the deficit distribution; in the second section we give three upper bounds by using the technique of probability generating function and mathematical induction, among which the later one is always more precise than the previous one. The results obtained in this thesis give some supplements of current literature for the ordinary discrete time renewal risk model.