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. In the classical Wiener setting, this limit is an inverse Gaussian \quad  L\'evy process. By continuous mapping, analogous results follow for any functional, such as a derivative payoff or price, of our volatility paths. As an example, the classical Heston log-price limit is shown to be very closely related, but not equivalent, to the normal-inverse Gaussian L\'evy process. This exposes the origin of the marginal connection between these two processes revealed in Mechkov (2015) .
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) 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 , \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 subordinator .
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Our first main results are the existence and uniqueness of solutions to a `generalised CIR initial-value problem' , which originates from the classical
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
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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)
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
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thereby also accommodating discontinuous paths. In the classical Wiener setting, this limit is an inverse Gaussian
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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
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process. By continuous mapping, analogous results follow for any functional, such as a derivative payoff or price, of our volatility paths. As an example, the classical Heston log-price limit is shown to be very closely related, but not equivalent, to the normal-inverse Gaussian L\'evy process. This exposes the origin of the marginal connection between these two processes revealed in Mechkov (2015)
subordinator