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Pathwise volatility: Cox-Ingersoll-Ross initial-value problems and their fast reversion exit-time limits

Ryan McCrickerd

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Abstract: 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.

Date: 2019-02, Revised 2019-04
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