A comparison of four methods for simulating the diffusion process

Four methods for the simulation of the Wiener process with constant drift and variance are described. These four methods are (1) approximating the diffusion process by a random walk with very small time steps; (2) drawing directly from the joint density of responses and reaction time by means of a (possibly) repeated application of a rejection algorithm; (3) using a discrete approximation to the stochastic differential equation describing the diffusion process; and (4) a probability integral transform method approximating the inverse of the cumulative distribution function of the diffusion process. The four methods for simulating response probabilities and response times are compared on two criteria: simulation speed and accuracy of the simulation. It is concluded that the rejection-based and probability integral transform method perform best on both criteria, and that the stochastic differential approximation is worst. An important drawback of the rejection method is that it is applicable only to the Wiener process, whereas the probability integral transform method is more general.