log-log plot, then a poor statistics at the tails may mislead to think that the distribution is log-normal.

![PDF of normalized volatility of stocks traded in US equity market (data from Ref. [@micciche]). The full line corresponds to a fitting by the theoretical PDF given by expression (\[fitvol\]). Fitting parameters are ($q,\nu,v^2_o)\simeq(1.178,2.20,0.097)$. Insets: linear-linear and log-log representation of the same data. []{data-label="fig:data"}](vfigure3.eps){width="50.00000%"}

Underlying dynamics {#underlying-dynamics .unnumbered}
-------------------

The satisfactory agreement between the empirical data and Eq. (\[fitvol\]) suggests that processes similar to either [**A**]{} or [**B**]{} may rule squared-volatility evolution. Hence, let us look at the explicit form of the ILEs associated to processes [**A**]{} and [**B**]{}: $$\label{ILEA}
\mbox{\bf A:}\;\;\;\; {\rm d} x =-\gamma[x-\theta]\frac{1}{x}{\rm d}t + 
\mu \sqrt{x}{\rm d}z + \alpha {\rm d}w \,.$$ $$\label{ILEB}
\mbox{\bf B:}\;\;\; {\rm d} x =-\gamma[x-\theta]\frac{1}{x^2}{\rm d}t + 
\mu \frac{1}{\sqrt{x}}{\rm d}z + \alpha {\rm d}w \,.$$ The first term in each ILE represents the deterministic restoring force with respect to the level $\theta$. It derives from a confining potential of the form $\gamma[x-\theta \ln x]$ ([**A**]{}) or $\gamma[\ln x +\theta/x]$ ([**B**]{}). In both cases, the potential has a minimum located at $x=\theta$ and is divergent at $x=0$.

Average values are $$\label{averages}
\langle x \rangle_{\bf A} 