How Compound Poisson Processes Relate to Marked Poisson Processes
Summary
The document distinguishes a compound Poisson process, formed by summing random jump sizes up to the count of a Poisson process, from a marked Poisson process, which pairs event times with marks. A compound process can be constructed from a marked process by attaching random values to its Poisson events, but the two terms describe different objects: one is an accumulated sum, while the other represents marked points.
The response says compound Poisson jump sizes are typically nonnegative independent and identically distributed real values, whereas marks in a marked process can be broader, potentially history-dependent, and need not be real numbers. In such general cases, summing the marks may not be defined. The explanation is conceptual rather than a formal treatment of dependence assumptions or alternative definitions, so the stated distinction should be read in the scope described by the answer.
Key ideas
- A compound Poisson process sums jump values over the arrivals of a Poisson process.
- A marked Poisson process associates marks with event times and can serve as the construction behind a compound process.
- Marks can have broader forms and dependencies than the nonnegative independent jump values described for a compound process.
- A sum of marks is not generally meaningful when marks are not real-valued.
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# Marked poisson process vs compounded
# Marked poisson process vs compounded
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} \}$ ?
## Answer by mchen (score 2)
https://quant.stackexchange.com/a/69896
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.