Before revision: Our first main results are the existence and uniqueness of solutions to a `generalised CIR initial-value problem' , which originates from the classical Cox-Ingersoll-Ross (CIR) stochastic differential equation after a time-change. These results hold on a pathwise basis for any continuous driver, and solutions are shown to accommodate any practicable continuous volatility path. We then prove a certain `fast reversion'pathwise convergence of our solutions to a hitting-time limit on the (D,M) topologies of Skorokhod (1956) , thereby also accommodating discontinuous paths.
After revision: Our first main results are the existence and uniqueness of solutions to a `generalised CIR initial-value problem' , which originates from the classical Cox-Ingersoll-Ross (CIR) processes, but in contrast prove well-defined without reference to a probability measure. This provides an unusual example of Skorokhod's representation theorem. In particular, we become able to generalise Heston's model of volatility to an arbitrary degree, by sampling drivers \omega under any probability measure; a rough one if so desired. Our main analysis relates to separable initial-value problems of type x'= \omega(x) + t - x + 1 , thereby also accommodating discontinuous paths.