by the multiplicative contribution (with a small value of parameter $s$).

In early works, only very particular values of $(r,s)$ have been considered. However, this may be sometimes owed more to reasons of mathematical solvability, than to econophysical ones. Following the above discussion, $(r,s)$ may be non-universal, depending on the particular nature of a market or its agents. Therefore, we will not discard any possibility [*a priori*]{}.

Generalized multiplicative process with mean reversion {#Sec:mean_rev}
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We consider the simple class of stochastic multiplicative differential equations given by Eq. (\[ILE0\]), that generalizes many processes usually found in the literature of volatilities. We investigate, in this Section, the long-time PDFs that this class of processes yields. The Fokker-Planck equation associated to Eq. (\[ILE0\]), following standard methods[@books], is $$\label{FP0}
\partial_t\rho 
= \gamma \partial_x( [x-\theta]x^{r-1}\rho) + \mu^2 \partial^2_{xx} [x^{2s}\rho] \;.$$ Its long-term solution is relevant in connection to the assumption that the process can be treated as quasi-stationary. In that case the PDF obtained from an actual data series will coincide with the stationary solution. Considering reflecting boundary conditions at $x=0$ and $x\to\infty$[@books], the steady state solution of Eq. (\[FP0\]) reads:

$$\label{ss0}
\rho(x)=\frac{\rho_o}{x^{2s}}  \exp\left( -\gamma_\mu\Bigl[
\frac{x^{p+1}}{p+1} - \theta\frac{x^{p}}{p} \Bigr]\right),$$

with $p\equiv r-2s\neq0,-1$, where $\rho_o$ 