Before revision:  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.
After revision:  After bounding these solutions in finance, such as an equity price or foreign exchange rate. This can be related to the conventional Ito calculus-based framework through the time integral of a price's squared volatility. In the new framework, corresponding processes are strictly increasing, solve random ODEs, and are composed with geometric Brownian motion to obtain price processes. The new framework has no dependence on stochastic calculus, so processes can be studied on a pathwise basis using probability-free ODE techniques and functional analysis. The ODEs considered depend on continuous driving functions which are `spatially irregular' , we prove a rare type of convergence result, towards c\`adl\`ag exit-time limits, on Skorokhod' s M_1 topology.