Optimal Investment And Ruin Probabilities For An Insurer In Stochastic Environment

With the development of financial markets, insurance companies face a lot of competition and challenges. And business profits simply relying on the traditional underwriting reduce gradually. Therefore, the insurance companies will make investment as an important aspect. Investment has become a pillar for the survival and development of an insurance company. Classical risk model has not simulated the realistic situation of risk. So the model with investment has become a hot issue in the field of the modern actuarial science and mathematics. Also, because both of the amounts of premium incomes in different seasons and the claim periods are different, the premiums, claims and investment returns are different. And constant premium rates will not meet the actual demands. To solve this problem, this paper focuses on the model of investment with variable proportion and the premium controlled by the Markov chain, which claims process is a compound Poisson process. The main contents and achievements are as follows:1. At first, we introduce the background of risk theory and its development. Then, classical risk model, its promotion and the relevant conclusions are introduced. And we present the contents and structure of this paper.2. We give the risk model with random environment and the investment of variable proportion. And we conclude the Bellman equation of survival probability, using It?'s formula.3. We consider the optimal investment strategy of maximal survival probability, with studying the parameters of the Bellman equation.4. We conclude the integral differential equation of survival probability, and prove the existence of solution. Then, we obtain the numerical solution of the integral differential equation.