We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

我们为带有负相位类型跳跃的Lévy过程提供了尺度函数的新颖表达。它是根据某个跃迁速率矩阵确定的，该矩阵显式地直到单个正数。提出了一种用于后者计算的单调迭代方案，结果表明误差以指数方式快速衰减。我们的数值示例表明，该算法允许我们采用具有数百个相位的相位类型分布，这在将已知公式用于根函数的比例函数时是有问题的。可以预期扩展到其他分布，例如矩阵指数和无限维相位类型。