Time Index of Lévy Process

Question

Consider (for all $t\geq 0$) a linear time transformation function $\nu(t)=at+b$ with the following properties:

1. $\nu(0)=-1$
2. $\nu(t)$ is an increasing function of the time index $t$ i.e. $a>0$.
3. $\nu(t)$ is a continuous function of the time index $t$.
4. $\nu(t)$ is a 1-1 function of time index $t$.
5. $\nu(t+1)$-$\nu(t)=a$ for all $t\geq 0$.

Q: Can we define a Lévy process with $\nu(t)$ as the time index (rather than $t$) or do we need more conditions on $\nu(t)$? In other words, I would like to see the possibility of defining $\{Z_{\nu(t)};\nu(t)\geq -1\}$, where $Z_{\nu(t)}$ is a Lévy process.

P.S. I originally asked this question on http://stats.stackexchange.com, but I couldn't find an answer, so I posted it here.