According to the computation of fphy-03-00011 the position for process satisfying has the fourth moment
This is a situation where kurtosis can be growing with 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 denote the fundamental solution of the fractional diffusion with a constant . The Fourier transform is (see fphy-03-00011). The computation of the kurtosis comes from