Several Results On Risk And Random Networks

In this paper, we concern about several problems on insurance risk and randomnetworks. In the first part, we first present several properties of heavy-tailed distributions. Thenthe stationary renewal risk model with constant interest force is discussed. We present therelation formula between ψ_δ (u) and ψ~0_δ (u), where ψ_δ (u) is the ruin probability in thestationary renewal risk model with constant interest force; ψ~0_δ (u) is the ruin probabilityin the renewal risk model with constant interest force. We get the series expansions ofψ_δ (u) and the (joint) distributions of several important actuarial diagnostics. In practice,insurance companies seek to reduce the risk of ruin by means of reinsurances. We considertwo types of reinsurances respectively and correct the classical risk model in this part.Then the Lundberg inequalities and the formulas are concluded through stochasticprocesses and martingale theory. In the second part, in order to explore further the mechanism responsible for scale-freenetworks, we introduce two extended models of the Barabási –Albert model. The two mod-els give more realistic descriptions of the local processes than the Barabási –Albert model,incorporating the addition of new nodes, new links between old nodes, the rewiring and de-letion of some links. We prove that the two extended models are models of scale-free netw-orks if the parameters of the models are choosen properly and get the scaling exponent γby the continuum theory.