Q:

Why does jump process has to be Cadlag and not the other way around

In all books and references that I have been exposed to, the jump processes have been defined to be Cadlag(right continuous with left limits). But no one has explained why this is the preferable case, why can't it be Caglad? 
I suspect it has something to do with filtration, but I don't know the exact reasoning. 

A:

I don't know if this is enough. But here is my understanding.
Let's imagine a simple process like a Poisson process.
It is naturally cadlag, because at the time you jump, you jump. Just before, you have not jumped. Mathematically, if the first jump occurs at $t$, $\forall s<t, N_s=0$ and $N_t=1$. It means that the jump occuring at time $t$ is $t$-measurable (even if it is not predictible).
So a cadlag process means that at the time of the jump, you see the process jumping.

