On First Crossing Time Densities of Normal Processes with Oscillatory Covariances

A vector algorithm is provided for simulating stochastic stationary normal processes with rational spectral densities, in order to determine first-crossing-time densities for the general case of time-varying boundaries. The results of massive simulations are discussed and conclusions are drawn on the effects of boundaries and covariances oscillations on determining qualitative and quantitative features of the densities. The observed shapes are seen to depend not only on the forms of the boundaries but also on the specific numerical values of the parameters of the processes. Indeed, they turn out to be unimodal, bimodal or multimodal even for fixed shapes of the covariance functions and of the boundaries. This is unexpectedly different from the case of diffusion processes whose first-crossing-time densities are either unimodal, or rigorously multimodal in the case of periodic boundaries. Such a feature makes normal processes particular suitable for modeling purposes.