Although the ruin probability in a renewal insurance risk model with credit interest may be viewed as a classical research problem, exact solutions are only available in the literature in very special cases when both the claim amounts and the interclaim times follow distributions such as the exponential. This is a long standing problem especially from a computational point of view, and the difficulty lies in the fact that the ruin probability usually satisfies a higher order integro-differential equation and/or an ordinary differential equation with non-constant coefficients. In this paper, for a large class of interclaim time distributions (including a combination of exponentials), we shall develop an approximation for the ruin probability using Laguerre series expansion as a function of the initial surplus level. It is shown that the (approximated) Laguerre coefficients can be solved from a system of linear equations, a procedure that is very easy to implement. A main advantage of our approach is that no specific distributional assumption on the claim amounts is required, apart from some mild differentiability and integrability conditions that can be verified. Numerical examples are provided to illustrate the very good performance of our approximation including both light-tailed and heavy-tailed claims.

尽管具有信用利息的续保风险模型中的破产概率可能被视为一个经典的研究问题，但只有在索赔金额和索赔间隔时间都遵循指数分布等非常特殊的情况下，才能在文献中找到精确解。这是一个长期存在的问题，尤其是从计算的角度来看，困难在于破产概率通常满足高阶积分微分方程和/或具有非常数系数的常微分方程。在本文中，对于一大类索赔间时间分布（包括指数组合），我们将使用拉盖尔级数展开作为初始剩余水平的函数来开发破产概率的近似值。结果表明，（近似的）拉盖尔系数可以从线性方程组求解，这是一个非常容易实现的过程。我们的方法的一个主要优点是，除了可以验证的一些温和的可微性和可积分性条件外，不需要对索赔金额进行特定的分布假设。提供了数值示例来说明我们的近似值的非常好的性能，包括轻尾和重尾声明。