| Recently, the risk theory has developed very rapidly, many scholars have committed to studying the probability of ruin of insurance company, they put forward many new risk models, and these studies have enriched and developed the risk theory. Most of the scholars on the researches of the probability of ruin are mainly focused on the time to ruin, the surplus prior to ruin and the deficits until ruin, while few scholars study on the researches of relationship between the number of claims and the time to run. Therefore, it is innovative and meaningful to study the number of claim which is together with the time to ruin, and some known results relating to the time of claim can be interpreted in terms of the number of claim. There are certain theoretical values and practical significances to study the probability of ruin under the nth claim with different of risk models.Firstly, this paper narrates some basic knowledge and methods on the risk theory. Secondly, in order to make the risk model conform to the actual situation, on the basis of classical risk model, this paper studies the risk events and claims which may not be equivalent, namely compound Poisson-Geometric risk model. By constructing a Gerber-Shiu function, we derive the renewal equation of the Gerber-Shiu function, obtain the joint probability density function of the time to ruin and the number of claims until ruin, and find the formula of ruin probability when ruin occurs at the time of the nth claim. As well, considering in the classical risk model, the insurance company only faces the risk of a single situations, but in real life, the insurance company will take different insurance businesses. So, this paper also studies the problems of ruin on dual independent risk model and dual correlative claim risk model, and finally finds the formula of ruin probability when ruin occurs at the time of the nth claim.Finally, this study makes a conclusion and gives the prospect on the study. |
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