How to recognize that a Poisson Process that is split has two separate Poisson Process through viewing it as a filtration? I read in a paper that if we have a Poisson Process with rate $\lambda$, and that each arrival is separated into two separate categories, $A$ and $B$ with probability $p$ and $1-p$, then each of the two categories are themselves an independent Poisson Process with rate $\lambda p$ and $\lambda (1-p)$. However, the paper I read said that this is easily seen without proof as each of the two processes are just a filtration of the overall process. I am not sure why this is the case. Does anyone know?

The fact that this results in a Poisson process can be obtained without relying on filtering. Just consider that each of the two resulting processes have the Poisson properties (independence of occurences, stable in time, memoryless) and are therefore Poisson processes. The rates are easy to obtain, as a proportion $p$ of $\lambda$ goes to the first process, that therefore has a rate of $p.\lambda$. 
