The Ruin Problem In The Poisson-Geometric Risk Models With Diffusion

Risk theory as an important part of insurance actuarial, the object of study are the stochastic model and ruin probability and so on which in insurance industry. In the classic risk model the policies arrive at a constant rate and claim number follows poisson distribution, the average is equal to the variance. But in fact, the policy number of insurance company in the unit time is a random variable, and the variance of the claim number is more than the average, the divergence is relatively bigger. Because of the uncertain factors such as the inflation and investment profit make surplus process of insurance company have obvious variation, Gerber(1970) analyzed and studied the uncertain factors for the first time in [15]. and described the influence of the uncertain factors with the Brown motion, studied the ruin problem of the classical risk model with diffusion firstly. Based on above evidences, this paper assume that police arrive with Poisson process, and the claim number follows Poisson-Geometric distribution, then establish a Poisson-Geometric risk model with constant interest rate and diffusion. Firstly, under the lemma and the preparing theorem, obtain the upper boundary of the ultimate ruin probability of the new model. Secondly, when the claim amount is heavy-tailed distribution, we get the upper and lower boundary of ultimate ruin probability. Lastly, we consider a new Poisson-Geometric risk model with diffusion in which the insurance policy arrive with Poisson process and the insurance premium is also a random variable. Analyze the ruin time of the risk model, and get the expression of ultimate ruin probability and the Lundberg upper boundary.