I have little knowledge on jump processes, so I am trying to make an analogous case of a normal stochastic process here but I my intuition says that I am doing something wrong
Suppose you have a stochastic process $(S_t)_t$ such that
$$dS_t = S_{t^-} r_t dt + (S_{t^-} - A_t) \sigma_t dW_t - \sum_k (S_t^{-} \beta_k + \alpha_k) \delta_{\tau_k}(t) dt$$
where $A_t, r_t$ are deterministic processes, $\tau_k$ are deterministic times, and $\alpha_k, \beta_k$ are real sequences.
I want to define the process
$$M_t = \frac{1}{t} \int_0^t S_u du$$
and calculate the generator of the process $(t, S_t, M_t)$. I suppose I know the generator of the process $(t, S_t)$. How the jumps of $S_t$ affect $M_t$?
Can I just add $\frac{S - M}{t} \frac{\partial}{\partial M}$ in generator of $(t, S_t)$ to find the generator of $(t, S_t, M_t)$ or do I need to account the jumps in some way?