With The Bankruptcy Of The Risk Of Interference Model Study

The classical risk model,which was put forward earlier than other models has been studied most thorough. But it's an ideal model because there are many disturbances in real life. So it's very necessary to research the risk model perturbed by diffusion. Gerber-Shiu discounted penalty function became a hot topic from 1998,when it was introduced in risk theories. In this paper, Gerber-Shiu discounted penalty function and joint distributions of some actuarial random variables are discussed.The article is divided into 5 sections.Chapter 1 is about the development and current situation of risk model.Chapter 2 introduces some related stochastic process theories.Chapter 3 discusses Gerber-Shiu discounted penalty function in the risk model perturbed by diffusion with Barrier dividend strategy. First, the integral equation of Gerber-Shiu discounted penalty function is obtained from the knowledge of Brownian motionwhere t₀≤(?),0≤a≤(?),On this basis, the properties of continuity and twice-continuous differentiability for m_s(u) are also got from the boundness of m_s(u) and continuous differentiability of the functions H, h and F.Chapter 4 is founded on chapter 3. In this part, we make a further research of the risk model perturbed by diffusion with constant interest rate under Barrier dividend strategy. First, we get the integral equation of Gerber-Shiu discounted penalty function from the knowledge of Brownian motion WhereThen the twice-continuous differentiability of Gerber-Shiu discounted penalty functionis discussed by the boundness of m_s(u) and continuous differentiability of the functions H, h and F.At last, we get its integro-differential equation by the use of Ito's formulaIn chapter 5, ruin problem of risk model perturbed by diffusion with stochastic premium rate is considered. Suppose the premium rate is any discrete random variableξ(ω), whose values are k_i, i = 1,2,…. Then the expression of the risk model isWhenξ(ω)=k_i,U_i(t)=u+k_it-(?)X_i+W(t) for t≥0 is the classical modelperturbed by diffusion.Then the exact expressions for actuarial diagnostics, such as the ruin probability, the distribution of extreme surplus before ruin, the joint distribution of the surplus immediately before ruin, the deficit at ruin and the ruin time, are concluded through stochastic process and probability theory methods.