It is a well known fact, that it is very difficult to estimate the drift of a Brownian motion with drift from looking at a single path over a finite interval $[0, T]$. Is it the case with Levy processes with jumps?

Assume we are given a Levy process with drift $\gamma$ and Levy measure $\nu$, with no Gaussian component. And we are given a single path over a finite interval $[0, T]$ of this process. Is it possible to determine $\gamma$ without knowing the parameters?

If the process is compound Poisson then it seems that the answer is YES - because there are intervals of strictly positive length without jumps, which we can use to read the drift. What is the answer for processes with infinite activity? Finite/infinite variation?