关键词： 对偶模型, 随机观察, 折罚函数, 微积分方程

Abstract: A dual risk model for the surplus process where the process can only be observed at random observation times is considered.Tthe Gerbershiu expected discounted penalty  function mδ(u) which satisfies the integro-differential equations is derived.Moreover,when the individual gains size distribution is exponential f(x)=ve-vx and penalty function ω(x)=e-r2x,the explicit expressions for  mδ(u) are obtained.

Key words: dual model, random observation, the expected discounted penalty function, integro-differential equation