This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone clarify what is the relationship between the jump measure, levy measure and if it is possible to express these quantities as a $$\sum_{\Delta X_t \neq 0} sth$$ in the context of Ito's lemma?

In particular I am looking at the application to the exponential function at the moment, if it simplifies anything.

EDIT: I am trying to link together the martingale representation theorem as in the answer to this question (Cont Tankov use the analogous notation): 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.