A new procedure to find the ultimate ruin probability in the Cramér-Lundberg risk model is presented for claims with a mixture of m Erlang distributions. The method requires to solve an m order linear recurrence sequence, which translates into finding the roots of an m-th degree polynomial and solving a system of m linear equations. We here study only the case when the roots of the polynomial are simple. A new approximation method for the ruin probability is also proposed based on this procedure and the simulation of a Poisson random variable. Several analytical expressions already known for the ruin probability in the case of Erlang claims, or mixtures of these, are recovered. Numerical results and plots from R programming are provided as examples.

针对混合了m 个Erlang 分布的索赔，提出了一种在 Cramér-Lundberg 风险模型中寻找最终破产概率的新程序。该方法需要解决的米阶线性递归序列，这转化为找到一个的树根米个多项式和解决的一个系统米线性方程。我们这里只研究多项式的根很简单的情况。基于该程序和泊松随机变量的模拟，还提出了一种新的破产概率近似方法。在 Erlang 索赔的情况下，已经为破产概率已知的几个分析表达式，或这些的混合，被恢复。R 的数值结果和绘图 编程作为示例提供。