EDIT:I am trying to link together the martingale representation theorem as in the answer to this question (Cont Tankov use the analogous notation): http://mathoverflow.net/questions/70981/martingale-representation-theorem-for-levy-processes
Under some integrability conditions, there exist $\phi$ and $\psi$ st:
$$M_t=M_0+\int^t_0 \phi(s)dW_s+\int^t_0 \int_{\textbf{R}}\psi(s,x)\tilde{N}(ds,dx)$$Where $\tilde{N}(ds,dx)$ is the compensated measure of the Lévy process $X$.
and the Ito's formula for Levy processes (taken from Cont Tankov) is:
$$f(X_t) - f(X_0) = \int_0^t \frac{\sigma^2}{2} f''(X_s) ds + \int_0^t f'(X_{s-}) dX_s$$$$ + \sum_{0\leq s \leq t} \textbf{1}_{(\Delta X_s \neq 0)}[f(X_{s}) - f(X_{s-}) - \Delta X_s f'(X_{s-})]$$
Please note that the formulas above are for scalar processes.
My question is how does the term $\int^t_0\int_{\textbf{R}}\psi(s,x)\tilde{N}(ds,dx)$ relate to $\sum_{0\leq s \leq t} \textbf{1}_{(\Delta X_s \neq 0)}[f(X_{s}) - f(X_{s-}) - \Delta X_s f'(X_{s-})]$ and to the Levy measure of the process $X$.
Also Proposition 8.16 from Cont-Tankov suggests that a funtion of a Levy process is a martinale iff we have:
$$f(X_t) - f(X_0) = \int_0^t f'(X_s)\sigma dWs + \int_{[0,t]\times R} \tilde{N}(ds,dx) [f(X_{s-} + y) - f(X_{s-})]$$
And again I am not sure how to relate the formula above to the Ito's formula.
