Fast and accurate calculations for first-passage times in Wiener diffusion models

We propose a new method for quickly calculating the probability density function for first passage times in simple Wiener diffusion models, extending an earlier method used by Van Zandt, Colonius and Proctor (2000). The method relies on the observation that there are two distinct infinite series expansions of this probability density, one of which converges quickly for small time values, while the other converges quickly at large time values. By deriving error bounds associated with finite truncation of either expansion, we are able to determine analytically which of the two versions should be applied in any particular context. The bounds indicate that, even for extremely stringent error tolerances, no more than 8 terms are required to calculate the probability density. By making the calculation of this distribution tractable, the goal is to allow more complex extensions of Wiener diffusion models to be developed.