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.
After revision: Motivated by successes of fast reverting volatility models, and the implicit dependence of `rough' processes on infinitesimal reversionary timescales, we establish a pathwise volatility framework which leads to a complete understanding of volatility trajectories' behaviour in the limit of infinitely-fast reversion. Towards this, we first establish processes that are weakly equivalent to Cox-Ingersoll-Ross (CIR) stochastic differential equation after a time-change.