A Study On Mathematical Risk Model Of Insurance

Risk theory is mainly that how to make use of the methods of probability and stochastic process to establish mathematical model, according to some actual situation about insurance company's management ,and analysis its surplus process to give some means to calculate premium, some results about ruin probability, adjusted coefficient and so on. Because insurance solvency is a very important problem focused on by insurer in reality, and ruin probability-bankrupt possibility of manager-can describe stability and insolvency of insurance company, the problem of ruin probability is straight the key subject in risk theory especially ruin theory. This paper have a lot of improvement from different angle in the foundation of classic risk model, so we obtain different risk model, give some expressions for the ultimate ruin probability in corresponding model. Among the discrete-time risk models, one mostly discussed by people is compound binomial model, the paper popularize it from the following aspects: on one hand because classic discrete-time risk model is single line model ,and it have certain limitations, the article give a new double-line risk model from reality after many literatures have read ,and separately use Poisson distribution, binomial distribution to simulate the claim amount, the claim number of double-type insurance ,apply martingale theory to get ruin probability. On the other hand, considered interest and investment factors, we give two kinds of discrete-time ruin model under the influence of two factors, separately discuss their ruin probability, and show the effect of two factors, generalize the relate results. Among the continuous-time risk models, the most classic one is compound Poisson model, founded by Lundberg in 1905.the article generalize it from the flowing three aspects: firstly we replace Poisson process with general stochastic dot process for example Cox process to describe the claim number. secondly we use more scientific method to simulate insurance premium, for instance: displace constant premium rate with Markov-modulated premium rate. thirdly considering the uncertain of insurance company's income ,we describe it with Brown motion So the article separately get many risk models, and we take use of stochastic theory to discuss the upper bound of ruin probability ,give some explicit expressions for ruin probability. At last ,the paper make some mends on negative risk model such as life pension , use Poisson to simulate insurance's income ,so we obtain a double Poisson processes risk model, prove Lundberg inequality for ruin probability with martingale theory ,also get a similar expression for it. Finally all the works in this thesis are summarized and some prospects are also proposed.