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Let $G(u) = u \frac{d}{du} E_{\alpha}(-u)$.  The innovations of the volatility process for equities should then be distributed as the inverse Fourier transform of this function modulo scaling.  This is a conjecture directly from the anomalous diffusion literature (e.g. metzler-klafter fphy-03-00011).  It is not difficult to verify that for these the kurtosis is an increasing function of the tail index as follows, after changes of variables and application of Fubini’s theorem a few times using the representation of the fundamental solution of $D_t^\alpha u = -D_{\alpha,2}\Delta$ as the inverse Fourier transform of a Mittag-Leffler function,
$\frac{d}{dt} \langle x^4(t) \rangle = C(D_{\alpha,2}, t, \alpha) \int du \int dy y^4 e^{-2\pi i yu} u \frac{d}{du} E_{\alpha}(-u)$
where $C(D_{\alpha,2}, t, \alpha)$ is increasing in $\alpha$