I have to prove that a Lévy process is a semimartingale.

In general we say that $X$ is a semimartingale if it is an adapted process such that, for each $t ≥ 0$,

$$X (t) = X (0) + M(t) + C(t)$$ where $M = (M(t), t ≥ 0)$ is a local martingale and $C = (C(t), t ≥ 0)$ is an adapted process of finite variation.

By the Lévy–Itô decomposition we have, for each $t ≥ 0$,

$X (t) = M(t) + C(t)$,

where $M(t) = B_A(t) + \int_{|x|\leq1} x \tilde{N} (t, dx)$ and $C(t) = bt +\int_{|x|>1} xN(t, dx)$, with $$\int_{A} xN(t,dx)= \sum_{0 \leq u \leq t} \Delta X_u 1_A(\Delta X_u)$$ and $$\int_A x\tilde{N}(t,dx):= \int_A xN(t,dx)-t\int_A x\nu(dx).$$

Now, I think I can prove that $M$ is a martingale and that $C$ is of finite variation, but with respect to which filtration?

Because the book "Lévy Processes and Stochastic Calculus" of Applebaum doesn't say anything about the filtration but in another book I read that $M$ and $C$ are to be adapted to the same filtration to which X is adapted.

But, for example I can prove that $(B_A(t),t\geq 0)$ is a martingale wrt his natural filtration, not wrt the filtration to which the Lévy processes $X$ is adapted.

Similarly, $\int_{|x|\leq1} x \tilde{N} (t, dx)$ is a martingale wrt his natural filtration?

Thanks.