On first-passage-time and transition densities for strongly symmetric diffusion processes

A subclass of symmetric one-dimensional diffusion processes is defined whose transition pdf’s satisfy some “strong” symmetry properties with respect to certain symmetry curves. The purpose is to make use of a variant of the method of images in order to determine a class of time-varying boundaries for which first-passage-time pdf and transition pdf with absorbing conditions on the boundaries can be obtained in closed form for the cases of a single boundary and of a pair of boundaries. New transition pdf’s in the presence of time-dependent boundaries as well as new first-passage-time pdf are thus disclosed. The practical usefulness of these results is pointed out via the derivation of first passage time pdf’s and transition pdf’s in the presence of nontrivial time-varying absorbing boundaries for the hyperbolic process and for the Ornstein-Uhlenbeck process.