In this paper we derive a new direct inversion method to simulate squared Bessel processes. Since the transition probability of these processes can be represented by a non-central chi-square distribution, we construct an efficient and accurate algorithm to simulate non-central chi-square variables. In this method, the dimension of the squared Bessel process, equivalently the degrees of freedom of the chi-square distribution, is treated as a variable. We therefore use a two-dimensional Chebyshev expansion to approximate the inverse function of the central chi-square distribution with one variable being the degrees of freedom. The method is accurate and efficient for any value of degrees of freedom including the computationally challenging case of small values. One advantage of the method is that noncentral chi-square samples can be generated for a whole range of values of degrees of freedom using the same Chebyshev coefficients. The squared Bessel process is a building block for the well-known Cox-Ingersoll-Ross (CIR) processes, which can be generated from squared Bessel processes through time change and linear transformation. Our direct inversion method thus allows the efficient and accurate simulation of these processes, which are used as models in a wide variety of applications.

翻译：本文提出了一种新的直接反演方法来模拟平方贝塞尔过程。由于这类过程的转移概率可由非中心卡方分布表示，我们构建了一种高效且精确的算法来模拟非中心卡方变量。在该方法中，平方贝塞尔过程的维度（等价于卡方分布的自由度）被视作变量处理。因此，我们采用二维切比雪夫展开来逼近中心卡方分布的反函数，其中一个变量即为自由度。该方法对任意自由度的取值均具有高精度与高效性，特别适用于计算难度较高的小自由度情形。本方法的优势在于：使用同一组切比雪夫系数即可生成涵盖整个自由度取值范围内的非中心卡方样本。平方贝塞尔过程是著名的Cox-Ingersoll-Ross（CIR）过程的基础构件，后者可通过时间变换与线性变换从平方贝塞尔过程导出。因此，我们的直接反演方法能够实现对这类过程的高效精确模拟，而此类过程在众多应用领域中均被广泛用作基础模型。