The Research Of Ruin Probability For The Several Risk Models Disturbed By Diffusion

Risk theory is one of the most important parts in the actuarial mathematics. It mainly deals with the insurance operation, It has been over 100 years for the research of risk theory since the beginning of 20^th century, The researchers take various extensions and investigations for the classical risk model and build series of new stochastic risk models to describe the management of the insurance company. They also put the theory of the Poisson process, Markov process, martingale and renewal process to the researches of the new stochastic risk models.Firstly, this paper deeply summarizes two classical methods that are applied in the researches of risk theory. Secondly, in order to describe the management of the insurance company more accurately, we consider three kinds of risk models that are disturbed by diffusion. In chapter 3, we analyze a discrete time risk model with diffusion. We use a stochastic process to describe the premium process; using Doob's stopping theorem and martingale methods to obtain accurate expressions for the ruin probability as well as the Lundberg inequality for the new risk model. Furthermore, by the concrete example, the relationship between the ruin probability, the initial surplus, the premium income and the claim amounts is discussed. And for the research of the second risk model disturbed by diffusion, we obtain the Laplace transform of the discounted joint distribution of the ruin time, the surplus before ruin and the deficit at ruin. At last we obtain the supremum estimation of the finite time ruin probability and the infinite time ruin probability in the third new risk model.