I have always implicitly thought that for a counting process $N_t$, defining the compound process $$\sum_{i=1}^{N_t} X_i,$$ where $X_i$ are i.i.d, was pretty much equivalent to constructing a stochastic integral $$\int_0^t X_s\,dN_s=\sum_{\tau_i<t}X_{\tau_i},$$ if $\tau_i$ are the jump times of $N_t$, for "some" process $X_t.$
Then, I realised that if $N_t$ is a simple Poisson process, then the compensator of the first one is $\lambda t \mathbb E[X_1]$, while for the second it is $\lambda \int_0^t X_s\,ds$. So I guess the two constructions must be different. Hence my question, is there a way to find the compensator of the compound process, for a general point process $N_t$ with intensity $\lambda_t$?