Approximating inverse cumulative distribution functions to produce approximate random variables

Abstract

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 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 speed ups 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 speed ups of a factor of 250 or more.