On the densities of certain bounded diffusion processes

A comprehensive outline is presented for obtaining the Laplace transforms of the transition probability density functions and of the first-passage-time densities for one-dimensional time-homogeneous diffusion processes in the presence of absorbing and/or reflecting boundaries. In particular, the Laplace transform of the transition probability density function in the presence of pairs of reflecting boundaries are explicitly obtained. Symmetric diffusion processes are then specifically considered and explicit closed-form relations are then obtained for the hyperbolic diffusion process in the presence of absorbing and/or reflecting boundaries. The special cases of the Brownian motion and of the Hongler process are finally analyzed