The Gerber-Shiu function is a classical research topic in actuarial science.However, exact solutions are only available in the literature for very specific cases where the claim amounts follow distributions such as the exponential distribution. This presents a longstanding challenge, particularly from a computational perspective. For the classical risk process in continuous time, the Gerber-Shiu discounted penalty function satisfies a class of Volterra integral equations. In this paper, we use the collocation method to compute the Gerber-Shiu function for risk model with interest. Our methodology demonstrates that the function can be expressed as a linear algebraic system, which is straightforward to implement. One major advantage of our approach is that it does not require any specific distributional assumptions on the claim amounts, except for mild differentiability and continuity conditions that can be easily verified. We also examine the convergence orders of the collocation method. Finally, we present several numerical examples to illustrate the desirable performance of our proposed method.

翻译：Gerber-Shiu函数是精算科学中的经典研究课题。然而，现有文献仅针对索赔额服从指数分布等特定情况给出了精确解。这构成了一个长期存在的挑战，尤其是从计算角度来看。在连续时间经典风险过程中，Gerber-Shiu贴现惩罚函数满足一类Volterra积分方程。本文采用配置法计算含利息风险模型中的Gerber-Shiu函数。我们的方法表明，该函数可表示为易于实现的线性代数系统。该方法的一大优势在于，除可轻松验证的温和可微性与连续性条件外，无需对索赔额分布作任何特定假设。我们还研究了配置法的收敛阶。最后，通过若干数值算例展示所提方法的优良性能。