On the evaluation of first-passage-time densities via nonsingular integral equations

The algorithm proposed by A. Buonocore, A. G. Nobile and L. M. Ricciardi [Adv. Appl. Probab. 19, 784-800 (1987; Zbl 0632.60079)] to evaluate first-passage-time p.d.f.’s through time-dependent boundary via a new second-kind Volterra integral equation is extended in two directions. On the one hand, the authors prove that the method for regularizing the kernel of the integral equation is valid not only for Wiener and Ornstein-Uhlenbeck processes but can also be used for a more general class of diffusion processes whose free transition p.d.f.’s are known; on the other hand, they show that the method can also be employed to make the kernel continuous if the diffusion process is restricted by a reflecting boundary and the transition p.d.f. in the presence of such boundary is known. Several examples (lognormal process, hyperbolic process, Wiener process with a reflecting boundary, processes with linear drift and linear infinitesimal variance) are thoroughly discussed along with some computational results.