If I am at time $t$ and I know that in the future, at time $t+h$ a process $X_s$ will jump by a random quantity, can $X_s$ be a Lévy process? ($X_s$ jumps before and after $t+h$ at random times)
If not what properties of Levy processes does it violate?
If yes how does the filtration of time $t$ relates to the filtration of time $t+h$? Shouldn't the knowledge of jump time $t+h$ be included only in the filtrations $\mathcal{F}_w$ for $w\geq t+h$?
No it's not a Lévy process because Lévy processes must have the property (which is almost equivalent to a definition(*)) to be continuous in probability, in other words almost surely at a time s (for example t+h) the paths of the process are continuous. As you know that the process is almost surly discontinuous your process is not Lévy.
For a reference look at the first chapter of Protter's book on stochastic calculus.
Best regards Edit : (*)Yo have to independent and stationary increments hypotheses to get a Lévy process.