The Research Of The Risk Models With Investment By The Stochastic Process And The Optimal Control Theory

This thesis is based upon the classic risk model and the extended risk models. By taking the surplus capital being invested to enhance the insurance payment level into account in insurance business, a type of new and practical risk model was proposed.For this type of new models with investment, we study it mainly from these ways:First, we study this kind of model's ruin probabilities with different distributions. 1. Considering the premiums and the claims follow the compound discrete Poisson distribution in this risk model, a double compound Poisson distribution risk model with investment and interfere was founded. After analyzing the proposed model, the stationary-increment properties of the profit process and the statistical character of the risk process were obtained. And the formula of the ultimate ruin probability and its Lundberg inequality were also derived. Especially for the first risk model the various trends of the upper bound of ruin probability were simulated analytically along with the changing of the investment, premium size and claim size. This clearly reflected the relations between the various and the ruin probability. The results have the important significance. 2. Considering the premiums and the claims follow the compound negative binomial distribution in this risk model. In this part, we study compound negative binomial risk model for single type-insurance with investment and Interfere, then analyze double type-insurance for same conditions, getting the similar conclusions to the above.Secondly, considering the risk model with the utility theory and the discount factor, it is more sensible to study this risk model that be named the filter risk model. In this model we consider not only the investment but also the utility theory and the discount factor. This hardly emerged in the other theses. The upper-bounds of ruin probability are obtained by the martingale method.Lastly, the mode of the investment was improved. The investment includes not only the riskless asset, but also the risky asset. By the improvement, the stock market theory was introduced into the risk theory. It is more practically according with the development of the finance. In this part, using the optimal control of the stochastic control theory, we set up the HJB function of this kind of risk model. We can obtain the optimal strategy of investment in order to make the ruin probability being least.