The Ultimate Ruin Probabilities Of Several Risk Models With Constant Interest

With the rapid development of the economic，the uncertainties and risks that peopleare facing in the daily life are increasing gradually, so do the researches of the risktheory, especially, the ruin probability is becoming one of the most important parts ofthe studies. Moreover, because of the turbulence of the international situation, more andmore factors are taking into the studies of the ruin probability, for example, theincreasing huge insurance policies make the reinsurance to be an important aspect; theresearchers realize the importance of the interest rates because of the instability ofinternational financial markets.The classical risk model is one of the most important models in the industry of theruin probability. We modify it with some kinds of methods according to the reality’sdemand. At the same time, considering the impacts of the constant interest rates and thereinsurance, we get some conclusions about ruin probability.The main results of this paper are summarized as follows：1. Considering the increasing huge insurance policies and uncertain risks in the reallife, we improve the classical risk model to meet the present situation of the insurancecompany, and it is more effective to maintain the stability of the insurance companybecause of the improved model. In addition, we discuss the impacts of the marketfactors and add the interest rate in the reinsurance risk model to make it to be furtherpromoted. Since then, we get the reinsurance risk model with the impact of the constantinterest rate.2. Starting from the surplus process, we discuss continues and discrete casesrespectively of the model above. For continues reinsurance risk model, the integralexpression of the ruin probability and its upper bound are obtained by using themethods of differential equations and recursive calculations; besides, we get the exactexpression of the ruin probability when the individual claim has an exponentialdistribution with quota reinsurance, and the expressions are similar to the conclusions ofthe classical risk models. But for the discrete reinsurance risk model, we use the way ofrecursive calculations to get the expression and the upper bound of the ruin probability,and we prove the conclusions by using the method of the induction.