that leads to the [*log-normal*]{} distribution.

For type I ($r=2s-1$), notice that the PDF decays as a power law, for large $x$, and goes to zero faster than power law, for vanishing $x$. The power-law exponent is controlled by $s$ and $\gamma_\mu$, that is, all the model parameters, except $\theta$ are involved. In the particular case $r=s=1$, one recovers the Hull-White process[@hw] $$\label{HW}
{\rm d} x \;=\;-\gamma[x-\theta] {\rm d}t + \mu x{\rm d}z \;.$$

In case II ($r=2s$), observe that the PDF has opposite behavior: it increases at the origin as a power law and decays exponentially for large $x$. All the model parameters, including $\theta$ are buried in the power-law exponent. In particular, if $r=2s=1$, one gets the Heston model[@heston] $$\label{H}
{\rm d} x \;=\;-\gamma[x-\theta]{\rm d}t + \mu \sqrt{x}{\rm d}z \;.$$ If $r=2s=2$, the geometric Ornstein-Uhlenbeck process is obtained $$\label{Geo}
{\rm d} x \;=\;-\gamma[x-\theta]x{\rm d}t + \mu x{\rm d}z \;.$$

Diverse other models proposed in the literature can also be thought as particular instances of our generalized model. For example, the one proposed by Micciché et al. [@micciche] is in correspondence with the Hull-White model (\[HW\]), with $x$ representing volatility $v$, whereas in the latter $x\equiv v^2$. Also a 