How to show that this is a martingale process?

Question

$X_1,X_2,...,X_n.. \ $ are independent and $X_n\sim Pois (n)$.

How can I show that $S_n=X_1+X_2+...+X_n-n(n+1)/2$ is a martingale with respect to the natural filtration?

Thanks in advance!

Answer 1

We can see that $S_{n+1} = S_n + X_{n+1} - (n+1)$ therefore the expectation is given by

$$E[S_{n+1}|S_n] = E[S_n|S_n] + E[X_{n+1}] - n-1 $$ $$=S_n + (n+1)-(n+1)$$ $$=S_n\qquad\qquad\qquad\qquad\,\,\,$$

Where we have used that $E[Pois(n)] = n$ to get the expecation of $X_{n+1}$. Note also, that $E[S_n] = 0$ but since we already know $S_n$, $E[S_n|S_n] = S_n$ (somewhat tautologically).