I have the following SDE:
$dX_t = X_{t-}(\mu dt + \sigma dN_{t}^{+} - \sigma dN_{t}^{-})$
The $N_t$'s both represent poisson processes with intensity $\lambda$ that are independent of one another.
Let's say I have a function, $g = g(t, X_t)$, and I want to use Ito's Lemma to find the SDE g satisfies. I know how Ito's lemma applies to jump processes and jump-diffusion, but all the examples I find deal with a single jump-process.
Intuitively, I think the dynamics should be:
$dg_t = [\partial_tg(t, X_{t}) + \mu X_{t-}\partial_xg(t, X_{t})]dt + [g(t, X_{t-}+\sigma X_{t-}) - g(t, X_{t-})]dN_t^+ + [g(t, X_{t-}-\sigma X_{t-}) - g(t, X_{t-})]dN_t^-$
Is this the correct extension of Ito's lemma?