First-passage-time densities for time-non-homogeneous diffusion processes

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al.