Let . 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 as the inverse Fourier transform of a Mittag-Leffler function,
where is increasing in
This restriction comes from my recent discovery of an almost linear empirical relation between these using around 1900 stocks (with outliers removed).