Define $Z(t)=N(t)−λt$ where $N(t)$ is a Poisson Counting Process. Now, I have already proved that $E[Z(t)|Fs] =Z(s)$
I am left to prove $E[|Z(t)|]<∞$ , which I am finding tricky. I have tried to use Hölder's inequality, the Triangle property and other stuff...
I have reached the terms $2λt$ and $√λt$, but I am not sure how to prove that these terms are finite..
What if the parameter $λ$ is infinite? Any help is appreciated!
Triangle inequality states that $|Z(t)|=|N(t)+\lambda t|\leq |N(t)|+|\lambda t|=N(t)+t\lambda$. Now take expectations and you directly get that $E(|Z(t)|)\leq \lambda t +\lambda t =2\lambda t$ thats it.
Also , does it really make any sense to consider an infinite rate Poisson process? That is , if the first arrival takes place at an average of infinite amount of time, then is it really a "process". Suppose you open a Sandwich shop and customers arrive after an infinite amount of time, then you might as well close your shop and start a new business.
So, ALWAYS assume the rate of a Poisson Process to be finite .
As for why it's a martingale, what you need to do is use the independent increments property and stationary distribution property.
i.e. $N(t)-\lambda t = N(t)-N(s)+N(s)-\lambda t$ and $N(t)-N(s)$ is independent of $\mathcal{F}_{s}=\sigma(N(r):r\leq s)$ .
So $E(N(t)-\lambda t|\mathcal{F}_{s})=E(N(t)-N(s)|\mathcal{F}_{s})+E(N(s)|\mathcal{F}_{s})-\lambda t$
Use independence and that $N(s)$ is $\mathcal{F}_{s}$ measurable to get
\begin{align}&=E(N(t)-N(s))+N(s)-\lambda t\\ &=E(N(t-s))+N(s)-\lambda t\\ &=\lambda(t-s)+N(s)-\lambda t\\ &=N(s)-\lambda s\end{align}