will use $$\label{intmarg}
\int^x  \frac{y^{-1}}{1+\lambda^2y^{2s}}{\rm d}y\;\equiv\;
\ln x\,-\,\ln(1+\lambda^2x^{2s})/(2s).$$ By means of these definitions and their asymptotic formulas[@abram; @formula], we obtain the possible PDF shapes, in $(s,r)$-space, as schematized in Fig. \[fig:am\]. The marginal cases $r=0$ and $r=-1$ will be considered latter. In general, sufficiently large positive $s$ is required in order to yield power-law tails, otherwise, stretched exponential tails prevail, as for the processes considered in Sect. \[Sec:mean\_rev\]. The additive noise does not add new domains with power-law tails, although regions with stretched exponential law are excluded or included by the normalization condition. For vanishing $x$, the main difference with purely multiplicative processes is that, for positive both $s$ and $r$, the PDF is truncated at the origin. Notice that, as the PDF is finite at the origin, then, if $x$ is identified with the squared volatility ($x\equiv v^2$), the PDF for $v$ increases linearly at the origin.

Let us analyze, in more detail, the marginal cases $r=0$ and $r=-1$ that can yield power-laws in both asymptotic limits. From Eqs. (\[FPss\])-(\[intmarg\]), we obtain

[**A**]{}: For $r=0$, the PDF has the form $$\label{IIm}
\rho(x) = \rho_o  
\frac{x^{\gamma_\alpha\theta} \Theta(x)}
{[1+\lambda^2x^{2s}]^{\gamma_\alpha\theta/(2s)+1} }  \; ,$$ where $\Theta(x)\equiv\exp[-\gamma_\alpha x
\,_2F_1({\scriptstyle \frac{1}{2s},1,\frac{1}{2s}+1},-\lambda^2x^{2s})]$ is a smooth function 