Poisson Risk Model Perturbed by Diffusion Gerber-Shiu Function in the Compound under a Barrier Strategy

,bd u  , considering a classical compound poisson risk model perturbed by diffusion in the presence of a constant dividend. The integro expression of the Gerber-Shiu function is derivated by the strong markov property and also   , bd u  is continuous and twice continuously differentiable. then we obtain the integro-differential equation of the Gerber-Shiu function by ˆ Ito formula, It is unique to research the ultimate ruin probability due to oscillation compared with other articles. Finally, we give the explicit expression of solution of integro-differential equation satisfied by   , bd u  when the claim sizes are exponential distribution.