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KURTOSIS OF FRACTIONAL DIFFUSION: CONJECTURE OF VOLATILITY AS FRACTIONAL DIFFUSION

According to the computation of fphy-03-00011 the position $x(t)$ for process satisfying $D_t^\alpha = D_{\alpha,1} \Delta$ has the fourth moment

$\langle x^4(t) \rangle = \frac{24 D_{\alpha,1}^2}{\Gamma(2\alpha+1)} t^{2\alpha}$

This is a situation where kurtosis can be growing with $\alpha$ which makes this setting quite interesting for volatility modeling where I discovered yesterday a clear relation between tail index and kurtosis.  Here the graph from 1900 stocks after removal of outliers.

In this case ‘heavy-tailed’ volatility jump distributions seem likely to be modeled by fractional diffusions.

Let $P(t,x)$ denote the fundamental solution of the fractional diffusion $D_t^\alpha P(t,x) = D_{\alpha,2} \Delta P(t,x)$ with a constant $D_{\alpha,2}$.  The Fourier transform is $p(q,t) = E_{\alpha} ( - 2 D_{\alpha,2} q^2 t^\alpha )$ (see fphy-03-00011).  The computation of the kurtosis comes from

$\frac{d}{dt} \langle x^4(t) \rangle = \int_{\mathbf{R}} dx x^4 \frac{d}{dt} p(t,x) = \int dx x^4 F^{-1} \frac{d}{dt} E_{\alpha}(-D_{\alpha,2} q^2 t^{\alpha} )$