I would like to compute the infinitesimal generator of a Poisson process $N$ with intensity $\lambda$.
So I can write:
$$\mathbb{E}[\ f(N_{t+s})-f(N_s)\ |\ \mathcal{F_t^0} \ ] = \mathbb{E}[\ f(N_{t+s}-N_s+N_s)-f(N_s)\ |\ \mathcal{F_t^0} \ ]$$
$$= \sum_{k=0}^\infty \frac{e^{-t \lambda}(\lambda t)^k}{k!}\cdot(\ f(N_{t}+k)-f(N_t)\ ) $$
where $\mathcal{F_t^0}$ is the raw filtration generated by $N$ and $f$ is a bounded and measurable function.
The result of the series should be:
$$\lambda t e^{-\lambda t}[\ f(N_{t}+1)-f(N_t)\ ] +O(t^2)$$
but I do not see how. I tried to manipulate the series as it is done for the computation of the expected value of a Poisson random variable but did not get anywhere.
Best Answer
Write
$$\sum_{k=0}^{\infty} \frac{e^{-\lambda t} (\lambda t)^k}{k!} (f(N_t+k)-f(N_t)) = I_1+I_2+I_3$$
where
\begin{align*} I_1 &:= e^{-\lambda t} \cdot 1 \cdot (f(N_t+0)-f(N_t))=0, \\ I_2 &:= e^{-\lambda t} \lambda t (f(N_t+1)-f(N_t)) \end{align*}
and
$$I_3 := \sum_{k \geq 2} e^{-\lambda t} \frac{(\lambda t)^k}{k!} (f(N_t+k)-f(N_t)).$$
It remains to show that $|I_3| \leq O(t^2)$. To this end, we note that
$$\begin{align*} |I_3| &\leq 2 \|f\|_{\infty} e^{-\lambda t} \sum_{k \geq 2} \frac{(\lambda t)^k}{k!} \\ &= 2 \|f\|_{\infty} e^{-\lambda t} (e^{\lambda t}-1-\lambda t) \\ &\leq 2 \lambda^2 \|f\|_{\infty} t^2. \end{align*}$$
(For the last step use e.g. Taylor's formula.)