Ruin Probability For Risk Model Under Stochastic Environment & Stochastic Process In Complex Networks

The study of the probability of ruin has been the core content of actuarial risk theory. In recent years, there has been an increasing attention in the utilization of stochastic control theory to insurance related problems, but the classical risk model has a lot of restrictions with the reality of the insurance companies. The model with investment has become a hot issue in the field of the modern actuarial science and mathematics. The study of the risk model with varying premiums rate which fits the fact more precisely is also arising. In this paper, we build a compound Poisson risk model with investment under stochastic environment, the premium rate varies with the level of risk reserves, and the insurer invest with a constant proportion. We investigate the integro-differential equation for the probability of ruin and discuss the existence of the solution for the equation. We will consider an example for the numerical solution at the end.Considering that the continuous time can't be realized in the practical application, we study the the compound Pascal model and its nature under stochastic environment, get the upper-bound estimates of ruin probability in the compound Pascal risk Model with random interest under stochastic environment.Complex networks describe a wide range of systems in nature and society, the complex networks theory is permeating to many different subjects such as mathematics,biology and engineering etc. Stochastic process theory plays an important role in the research about the topology and characteristic of complex networks. Meanwhile, more and more networks in finance have been given in recent years. In this paper, we consider the stochastic process application in complex networks. We build the evolving model of scale-free networks with intrinsic links and discuss his scale-free nature. Then, we try to give the clusteing coefficient of the graph processes with nonlinear preferential attachment using a method based on the G-value Markov process.