Differential Equations for Monte Carlo Recycling and a GPU-Optimized Normal Quantile

This article presents differential equations and solution methods for the functions of the form $A(z) = F^{-1}(G(z))$, where $F$ and $G$ are cumulative distribution functions. Such functions allow the direct recycling of samples from one distribution into samples from another. The method may be developed analytically for certain special cases, and illuminate the idea that it is a more precise form of the traditional Cornish-Fisher expansion. In this manner the model risk of distributional risk may be assessed free of the Monte Carlo noise associated with resampling. The method may also be regarded as providing both analytical and numerical bases for doing more precise Cornish-Fisher transformations. Examples are given of equations for converting normal samples to Student t, and converting exponential to hyperbolic, variance gamma and normal. In the case of the normal distribution, the change of variables employed allows the sampling to take place to good accuracy based on a single rational approximation over a very wide range of the sample space. The avoidance of any branching statement is of use in optimal GPU computations, and we give example of branch-free normal quantiles that offer performance improvements in a GPU environment, while retaining the precision characteristics of well-known methods.