Before revision:  In the classical Wiener setting, this limit is an inverse Gaussian \quad  L\'evy process.
After revision:  In the classical Wiener setting, this limit is an inverse Gaussian \quad x(0)=0, with \omega only assumed continuous (not Lipschitz, nor H\"older), solutions of which \varphi correspond to time-averages of volatility trajectories \varphi'. Such solutions are shown to exist, be unique and bijective for any \omega, essentially placing no constraints on corresponding volatility trajectories, except for their non-negativity. After bounding these solutions in time, we prove a rare type of convergence result, towards c\`adl\`ag exit-time limits, on Skorokhod's M_1 topology. One immediate corollary of this limiting result is a weak connection between the time-averaged CIR process and the inverse-Gaussian L\'evy process.