The Risk Model Perturbed By Diffusion In An Economic Environment

The classical risk model was first brought forward by the Sweden actuary Lundberg in his doctorial thesis in 1903. Then the Sweden actuary Cramer made the model to base on the strict theory of the stochastic process. So the classical risk model is also called the Lundberg-Cramer model. The classical risk model has been studied more than one hundred years. At present it is nearly perfect and the exact calculated results for all the actuarial diagnostics are derived in closed form. With the development of risk model, the invest profit about diffusion and income of premium is very important for insurance company to keep the finance stable.In chapter 1, we present the development of the risk model and some improvement on the classical risk model and some research methods.In chapter 2, we give a model about the multi-type risk model of double Poisson. And discuss the ruin probability and the Lundberg inequality, using the "Martingale approach".In chapter 3, we research the models with interest. We have derived the explicit expressions of some actuarial diagnostics. This actuarial diagnostics include the ruin probability, the surplus prior to ruin, the supreme profits before ruin, the time that ruin last, the deficit that is more than M, and the distribution of the surplus prior to ruin, the deficit and the supreme profit before ruin. Motivated by Cai, we consider the risk model that the interest is a Markov chain, and derive some actuarial diagnostics.In chapter 4, consider the risk model perturbed by diffusion in an economic environment, we consider the ordinary risk model that the premium is not a constant in per-time. Using the heavy-tailed method and the quality of Brown, we got the upper and down bound for ruin probability.