is a normalization constant and $\gamma_\mu\equiv\gamma/\mu^2$ an effective restoring amplitude, such that $\gamma$ (a parameter associated to order) becomes reduced by the amplitude of multiplicative noise (associated to disorder).

![ Diagram of the asymptotic behavior of the PDF given by Eq. (\[ss0\]), in ($s,r$)-plane. Unshadowed regions and dotted borders identify regions excluded by the normalization condition. At the positive $r$-axis, the PDF is finite at the origin. Tilted lines denote the marginal cases $r=2s$ ($p=0$), with pure exponential tail and power-law growth at the origin, and $r=2s-1$ ($p=-1$), with power-law tail and exponential of $1/x$ growth at the origin (the threshold points of these lines have coordinates $([1+\gamma_\mu\theta]/2,0)$ and $([1-\gamma_\mu]/2,0)$, respectively). Parameter $a>0$, in the exponential formulas, as well as the power-law exponents, depend on model parameters. Symbols correspond to the special processes: Hull-White (HW), Heston (H), Ornstein-Ulhenbeck (OU) and geometric OU (GOU). []{data-label="fig:m"}](vfigure1.eps){width="40.00000%"}

The class of processes described by Eq. (\[ss0\]) thus generically yields asymptotic exponential-like behaviors for small or/and large values. As soon as $p+1>0$, a [*stretched exponential decay*]{} is obtained for large enough $x$, such that the argument of the exponential is proportional to $-x^{p+1}$. If $p<0$, a [*stretched exponential of the inverse argument*]{} ($-1/x^{|p|}$) is obtained 