\(dY_t=Z_{N_t} dN_t \) is a compound Poisson process with intensity \(\lambda\) and \( Z_{N_t} \) is a random variable for the jump size. \(dW_t \) is Brownian motion.

The jump diffusion process \(X_t\) is defined as

$$ dX_t=\nu_t dt+u_t dW_t + \eta_t dY_t$$

So the Ito's lemma for this jump diffusion process is

$$ df(X_t)=\nu_t f'(X_t)dt+u_t f'(X_t) dW_t+ \frac{1}{2}u^2_t f''(X_t)+(f(X_t)-f(X_t-))dN_t$$

The compensated Poisson process of \( dN_t \) is \( d\hat{N_t}=dN_t-\lambda dt \), which is a martingale. Then how to use \(d\hat{N_t} \) to make \( df(X_t) \) a martingale?

A simple subsitution as below seems somehow not correct:

$$ df(X_t)=\nu_t f'(X_t)dt+u_t f'(X_t) dW_t+ \frac{1}{2}u^2_t f''(X_t)+\lambda(f(X_t)-f(X_t-))dt+(f(X_t)-f(X_t-))d\hat{N_t}$$

According to page 26 of http://people.ucalgary.ca/~aswish/JumpProcesses.pdf, maybe it should be something like this:

$$ df(X_t)=\nu_t f'(X_t)dt+u_t f'(X_t) dW_t+ \frac{1}{2}u^2_t f''(X_t)+\lambda (E[(f(X_t)-f(X_t-)]-E[Z_{N_t}]\eta_tf'(X_t))dt+(f(X_t)-f(X_t-))d\hat{N_t}$$

but I can't understand why.