Study Some Classes Of Continuous Time Risk Models

In actuarial science,the usual models are usually based on the indepeneny assumption,that is,the premiums and claims are assumed to be two independent and identically distributed(iid) randon variables series, and diferent times of policies are independent of each other. But in the insurance company physically the management, counting process claims and premiums arrived at the counting process is dependent, and grows and presents a diversification near there is necessity to for this type of grow situation and provide more objective and actual risk model nearly. Addition, at the end of last century, the conception of the expected discounted penalty function at ruin was first introduced by Hans U. Gerber and Elias S. W. Shiu who are contemporary international leading experts at ruin theory. A number of particular cases of the expected discounted penalty function at ruin led to important quantities of interest in risk theory. The expected discounted penalty function at ruin being a powerful analytical tool made it possible to analyze the time of ruin, the surplus immediately before ruin, the deficit at ruin, and related quantities in a unified manner. In this thesis, we build up and study three classes of risk model:(1) We consider the survival probability problem of a double-type risk model in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is thinning process. A integral equation for the survival probability are gotten. The explicit formula of the survival probability for the infinite interval is obtained in the special case-exponential distribution.(2) We study the ruin probability problem of the double-type risk model perturbed in which the rate of premium income is regarded as a random variable, the arrival of insurance policies is a Poisson process and the process of claim occurring is thinning process. Using martingale method, the Lundberg inequality and the common formula for the ruin probability are proved.(3) To be consider for the initial state and initial distribution, premium rates markov process control by the risk model, by a backward differential argument, the integral equation satisfied by the expected discounted further more we obtain a recursive inequality about the expected discounted penalty with the stationary initial distribution and a simplified estimation of ruin probability with no initial reserve.