First-passage-time location function: Application to determine first-passage-time densities in diffusion processes

A time-dependent function, namely the First-Passage-Time Location function, is introduced in the context of the study of first-passage-times. From this function, a strategy is developed in order to solve numerically the Volterra integral equation of the second kind verified by the first-passage-time densities for diffusion processes. The proposed procedure provides the advantages in the application of quadrature methods in terms of an appropriate choice of the integration step, as well as an outstanding reduction in the computational cost. Some examples are developed showing the validity of that strategy as well as the computational advantages.