In this paper we analyze a method for approximating the first-passage time density and the corresponding distribution function for a CIR process. This approximation is obtained by truncating a series expansion involving the generalized Laguerre polynomials and the gamma probability density. The suggested approach involves a number of numerical issues which depend strongly on the coefficient of variation of the first passage time random variable. These issues are examined and solutions are proposed also involving the first passage time distribution function. Numerical results and comparisons with alternative approximation methods show the strengths and weaknesses of the proposed method. A general acceptance-rejection-like procedure, that makes use of the approximation, is presented. It allows the generation of first passage time data, even if its distribution is unknown.

翻译：本文分析了一种近似CIR过程首次通过时间密度函数及其对应分布函数的方法。该近似通过截断包含广义拉盖尔多项式与伽马概率密度的级数展开实现。所提出的方法涉及若干数值计算问题，这些问题强烈依赖于首次通过时间随机变量的变异系数。本文对这些数值问题进行了深入分析，并提出了包含首次通过时间分布函数在内的解决方案。数值实验及与替代近似方法的比较揭示了该方法的优劣。最后给出了一种基于接受-拒绝机制的通用流程，该流程利用本文提出的近似方法，能够在首次通过时间分布未知的情况下生成相应的数据样本。