I am a bit fuzzy about difference between compounded poisson process defined as $$\sum_{i=1}^{N_t} D_i $$ where $N_t$ is poisson process and $ D_i $ are iid random variables
and marked poisson process. Is compounded poisson a version of marked process $ \{(\tau_i, D_i), i \in \mathbb{N} \}$ ?
The compound Poisson process isn't technically a marked process because we formulate the process with respect to $\sum_i D_i$ instead of $(\tau_i, D_i)$. However the compound process is constructed from a marked process.
The compound Poisson point process or compound Poisson process is formed by adding random values or weights to each point of Poisson point process defined on some underlying space, so the process is constructed from a marked Poisson point process, where the marks form a collection of independent and identically distributed non-negative random variables.
https://en.wikipedia.org/wiki/Poisson_point_process#Compound_Poisson_point_process
The difference between a compound and marked Poisson process is that for a compound Poisson process the $D_i$ are non-negative iid but for a marked process the $D_i$ can depend all past history $(\tau_i, D_{i-1}, \tau_{i-1}, ...)$ and don't need to be non-negative or even a real number.
marks can be as diverse as integers, real numbers, lines, geometrical objects or other point processes.
https://en.wikipedia.org/wiki/Poisson_point_process#Marked_Poisson_point_process
As such, we can't really talk about $\sum_i D_i$ for a marked process because in general that sum isn't defined since $D_i$ might not even be a real number.
