基于正交伽马展开的CIR过程首达时间分布
Orthogonal gamma-based expansion for the CIR's first passage time distribution
AI总结：
本文研究CIR过程首达时间分布的近似问题，提出基于广义Laguerre多项式与伽马密度的截断级数展开法，分析数值问题并给出解决方案，结合实验验证方法优劣，还提出可生成未知分布首达时间数据的类接受-拒绝过程。
摘要：
本文分析了一种用于近似CIR过程首达时间密度及对应分布函数的方法。该近似通过截断包含广义Laguerre多项式与伽马概率密度的级数展开式得到。所提方法存在若干数值问题，其严重程度强烈依赖于首达时间随机变量的变异系数。本文对这些问题进行了研究，并提出了同时涉及首达时间分布函数的解决方案。数值结果及与其他替代近似方法的对比，展现了所提方法的优缺点。本文还提出了一种利用该近似的、类接受-拒绝的通用过程，即便首达时间的分布未知，也能生成首达时间数据。
英文摘要：
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.
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