The Problems Of The Stochastic Return And Debt Interest In Poisson Risk Model

Recent years, the Poisson risk model with stochastic return and diffusion have achieved fruitful results. Gerber and Yang (2007) studied the absolute ruin problems of the compound risk model with investment and diffusion. They also discussed the case when the claims were exponential distributed; Yin and Wen (2013) extended the model of Paulsen and Gjessing (1997a) and they made the claim sizes in (-∞,∞). They studied the moment generating functions and the dividend problems.In this paper, we still suppose the claim sizes and time of claim arrivals are indepen-dent. Two correlated Brown motions drive the insurance risk and investment return. When the surplus is negative the company can still be kept on by borrowing money from the bank through constant debt interest; when the surplus is none-negative, the company invests its money in two ways:the risky asset and risk-free asset; when the surplus reaches a level, the surplus is dividend to the customers.In this paper, we studied the risk model with investment income under absolute ruin, the independent stochastic variables representing the claims took values in (0,00), the article got integro-differential equations for Gerber-Shiu functions of the risk model, as it was not easy to get the correct solutions of the equations, the article changed the equations to the conditions of σ2/1α=2,λ=0,ω(x,y)=1: and Then their solutions were And then we discussed the expected discounted value of dividend payments of the barrier strategy and threshold strategy respectively, also the moment generating functions and got their integro-differential equations. As it was not easy to get the solutions, the article only discussed the problem in the special condition of p(x)=e-x, x€(0,∞),σ2/1a3=2,δ=b2=0,λ=a+c of the equation under barrier dividend strategy and σ2/1α=2,λ=0of the threshold dividend strategy.The rest of the paper was organized as follows. The main results were introduced in Section1; in Section2the risk model was derived; Section3derived the Geber-Shiu functions, boundary conditions and the solutions to the equations in special cases. Section4and5showed the integro-differential equations of the barrier and threshold dividend strategies, boundary conditions and some results of the equations in special conditions were obtained.