If \(N_t\) is a Poisson process with intensity \(\lambda \),and \(dX_t=\delta dN_t\), \(q_t=-N_t\), then the Ito's lemma for function \(H(X_t)\) should be

$$ dH(X_t)=[H(X_t+\delta)-H(X_t)]dN_t$$

For the function \(H(X_t,q_t)\), why it is not something like this?

$$ dH(X_t,q_t)=[H(X_t+\delta,q_t)-H(X_t,q_t)]dN_t+[H(X_t,q_t-1)-H(X_t,q_t)]dN_t$$

As I read from some book, it should be

$$ dH(X_t,q_t)=[H(X_t+\delta,q_t-1)-H(X_t,q_t)]dN_t$$

Can anyone give me a formal derivation of the last equation? I cannot find in a book. Thank you.