I'm a little confused about the definition (or rather the defining properties) of a Poisson process.
This is the definition my professor gave in class:
Definition Let $N = (N_t)_{t \geq 0}$ be a $\mathbb{N}_0$-valued process on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. We call $N$ a "Poisson process with intensity $\alpha$" ($\alpha > 0$) if:
- $N_0 = 0$ almost surely
- $N$ has independent increments
- $N$ has Poisson-distributed increments, i.e.: $$N_t - N_s \sim \text{Poiss}_{\alpha(t-s)} \quad \forall \ 0 \leq s < t$$
- $t \mapsto N_t$ is almost surely right-continuous and non-decreasing
But I also stumbled upon other definitions which require $N_0 = 0$ surely or the paths to be right-continuous (not only almost surely) or leaving out 4. completely.
So my question is: Are all these definitions of the Poisson process equivalent? And if no, why are there different definitions at all and which is the 'right' one?
To give some context: One of our exercises in class is to show that the time of the first jump $$W_1 \colon = \inf \{ t \in \mathbb{R}_+ \mid N_t \geq 1 \}$$ is measurable. My attempt was to write $\{ W_1 \leq s \} = \{ N_s \geq 1 \}$ for all $s \in \mathbb{R}_+$. But this only holds (if I'm not mistaken) if the paths are right-continuous surely. But by the definition we had in class, the paths are right-continuous only almost surely.
I did some research and have at least a partial answer to my question:
Regarding number 4. in the above definition, this apparently follows from 1. - 3.
As for my specific problem with the measurability of $W_1$, I came up with the following:
Let \begin{align*} M :&= \{ \omega \in \Omega \mid t \mapsto N_t(\omega) \textrm{ is not right-continuous in } s \textrm{ and } N_s(\omega) = 0 \} \\ &= \{ N_s = 0 \} \cap \bigcap_{k=1}^\infty \{N_{s+\frac{1}{k}} \geq 1\} \end{align*} Then we have $\{W_1 \leq s \} = \{N_s \geq 1\} \cup M$, which shows that $W_1$ is measurable.