Before revision:  Our main analysisrelates to separable initial-value problems of type x'= \omega(x) + t - x + 1, 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.
After revision:  Our main analysisrelates to separable initial-value problems of type x'= \omega(x) + t - x + 1, This thesis develops a new framework for modelling price processes in time , we prove a rare type of convergence result, towards c\`adl\`ag exit-time limits, on Skorokhod' s M_1 topology.