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中文摘要：
本文考虑了索赔时间间距为指数分布与Erlang（2）分布混合时的平均折现罚函数,建立了该函数所满足的积分-微分方程及更新方程,讨论了其Laplace解。最后得出了破产概率所满足的Beekman卷积公式及索赔量分布分别为Phase-type分布和Pareto分布时破产概率的明确表达式和近似表达式。
英文摘要：
In this paper, we consider a compound renewal risk process in which the claim inter-arrival times are the mixture of exponentials and Erlang（2） distributions. A Integrodifferential equation and a renewal equation for the expected discounted penalty function are derived, and we also discuss the Laplace transform of the function. Finally, we obtain the Beekman＇s convolution formula for the ruin probability and the explict solution and the asymptotic one in the case where the individenal claim amount distribution is Phase-type and Pareto, respectively.
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