On the Construction of a Special Class of Time-Inhomogeneous Diffusion Processes

A special class of time-inhomogeneous diffusion processes, generated starting from Gauss-Markov processes conditioned on the same initial state, is considered. This class includes many interesting diffusion processes with time-dependent infinitesimal drift and variance, for which the transition probability density function is explicitly determined. Moreover, closed form results for the first-passage-time density through suitable time-varying boundaries are obtained. Special cases, generated starting from Wiener and Ornstein–Uhlenbeck processes, are considered and widely discussed.