For random variables produced through the inverse transform method, approximate random variables are introduced, which are produced using approximations to a distribution’s inverse cumulative distribution function. These approximations are designed to be computationally inexpensive and much cheaper than library functions, which are exact to within machine precision and, thus, highly suitable for use in Monte Carlo simulations. The approximation errors they introduce can then be eliminated through use of the multilevel Monte Carlo method. Two approximations are presented for the Gaussian distribution: a piecewise constant on equally spaced intervals and a piecewise linear using geometrically decaying intervals. The errors of the approximations are bounded and the convergence demonstrated, and the computational savings are measured for C and C++ implementations. Implementations tailored for Intel and Arm hardware are inspected alongside hardware agnostic implementations built using OpenMP. The savings are incorporated into a nested multilevel Monte Carlo framework with the Euler-Maruyama scheme to exploit the speedups without losing accuracy, offering speed ups by a factor of 5–7. These ideas are empirically extended to the Milstein scheme and the non-central χ² distribution for the Cox-Ingersoll-Ross process, offering speedups of a factor of 250 or more.

对于通过逆变换方法产生的随机变量，引入了近似随机变量，这些变量是使用分布的逆累积分布函数的近似值产生的。这些近似值的设计成本低廉，比库函数便宜得多，库函数精确到机器精度范围内，因此非常适合在蒙特卡罗模拟中使用。然后可以通过使用多级蒙特卡罗方法来消除它们引入的近似误差。高斯分布有两种近似值：等间隔的分段常数和使用几何衰减间隔的分段线性。近似值的误差是有限的并且收敛性得到证明，计算节省是针对 C 和 C++ 实现进行测量的。针对 Intel 和 Arm 硬件定制的实现与使用 OpenMP 构建的硬件无关的实现一起进行检查。节省的成本被纳入采用 Euler-Maruyama 方案的嵌套多级蒙特卡洛框架中，以在不损失准确性的情况下利用加速，从而将速度提高 5-7 倍。这些想法根据经验扩展到 Milstein 方案和非中心 χ^{Cox-Ingersoll-Ross 过程的2}分布，可提供 250 倍或更多的加速。