While studying the Poisson process, I had a question about whether this process is differentiable in probability and whether it is differentiable in the mean square sense, i.e. $$E\left|\frac{N_{t+h} - N_t}{h} - \frac{dN_{t}}{dt}\right|^2 \rightarrow 0, h \rightarrow 0$$
But I find it difficult to answer this question because I can’t figure out where to start. I understand that $$N_{t+h} - N_t \xrightarrow{d} Poiss(\lambda h)$$ But what can be done next? I will be grateful for any hint.
Firstly, the second moment of that ratio diverges
$$E[N_{[a,b]}^{2}]=\lambda (b-a)(\lambda (b-a)+1),$$ where $N_{[a,b]}$ is the Poisson counting process in an interval $[a,b]$
and so
$$E\left[\frac{N_{[h,t+h]}^{2}}{h^{2}}\right]=\frac{\lambda(\lambda h+1)}{h}\to +\infty.$$
This implies that it cannot converge in $L^{2}$ because we have the contradiction
$$+\infty\leftarrow E[f_{n}^{2}]\leq E[(f_{n}-f)^{2}]+E[f^{2}]<\infty.$$
However, as mentioned in the comments, there is a stochastic calculus for Poisson point processes because their paths are right-continuous increasing and so one can just use the Riemann-Stieltjes integral see Revuz-Yor Chapter XII