Evaluation of upcrossing first passage time densities for Gaussian processes via a simulation procedure

The upcrossing first passage time problem through a constant or a periodic boundary is considered for stationary Gaussian processes possessing rational spectral densities. Along the lines indicated in [Ricciardi-Sato (1986)], a series expansion of the upcrossing probability density function of the first passage time is considered and the first order of its partial sum is evaluated. Using a parallel procedure to simulate sample paths, estimates of the upcrossing probability density function of the first passage time are constructed. A comparison is provided with the numerical results obtained by approximating the simulated upcrossing probability density by the first order partial sum of a series expansion. It is finally shown that, starting from small times, the upcrossing probability density is susceptible of an excellent exponential approximation for a wide range of constant boundaries.