Analysis of reflected diffusions via an exponential time-based transformation

Let $X(t)$ be a time-homogeneous diffusion process with state-space $[0,+infty)$, where 0 is a reflecting or entrance endpoint, and let $Z$ denote a random variable that describes the process $X(t)$ evaluated at an exponentially distributed random time. We propose a method to obtain closed-form expressions for the conditional density and the mean of a new diffusion process $Y(t)$, with the same state-space and with the same infinitesimal variance, whose drift depends on the infinitesimal moments of $X(t)$ and on the hazard rate function of $Z$. This method also allows us to obtain the Laplace transform of the first-passage-time density of $Y(t)$ through a lower constant boundary. We then discuss the relation between $Y(t)$ and the process $X(t)$ subject to catastrophes, as well as the interpretation of $Y(t)$ as a diffusion in a decreasing potential. We study in detail some special cases concerning diffusion processes obtained when $X(t)$ is the Wiener, Ornstein-Uhlenbeck, Bessel and Rayleigh process.