Vectorized simulations of normal processes and the first passage time probability problem

In order to gather information on the shape of first-passage-time probability density functions for stationary normal processes and time-varying boundaries, a simulation procedure has been set up which is based on a method (J.N. Franklin, 1965) suitable to be vectorized and implemented on parallel computers. According to the shape of covariances and boundaries and depending on the numerical values of their parameters, different behaviours of the densities are disclosed: these can be unimodal or bimodal with two peaks, but they can also exhibit periodically spaced peaks when boundaries or covariance include a periodic component.