Risk Model Related Issues

Risk theory is an important branch of applied mathematics, an important theory of insurance mathematics, a hot topic in the present actuarial science and mathematics research. Ruin probability becomes the core content of risk theory, as it is a main measure for insurance risk. So it has important theoretical and practical significance for insurance company. In this dissertation, based on the classical risk model, we get some generalized risk models by extending the risk model from claim process, premium rate aspects. We mainly study the Erlang(2) risk model, the Cox risk model with variable premium and the Cox risk model perturbed by diffusion. It mainly constants the following results:1. A Sparre Andersen risk model with Erlang(2) interclaim times is considered. The double-Laplace transform of the finite time survival probability is derived. The explicit expression of the finite time survival probability is obtained, when the claim sizes are exponentially distributed using the complex inversion of the double transform and Residue Theorem. The ultimate ruin probability is got when the time t tends to infinity. Finally, the corresponding results are obtained when the model is perturbed by diffusion, and compared the results for the two cases.2. The expected discounted penalty function is investigated under the influence of a premium rate which varies according to the intensity of claims, and the occurrence of claims is described by a Cox process in the considered risk model. By a backward differential argument, we drive the integral equation satisfied by the expected discounted penalty function. The integral equations satisfied by ruin probability, the moments of the surplus immediately prior to ruin and the deficit at ruin are received. Finally the laplace transform of ruin probability when the claim size is exponentially distribution and the claim intensity is two states. The express of ruin probability is obtained for some exact number.3. The classical risk model is researched by the limit theory for random sums...