Let $(S_n)_{n\geq 0}$ be the simple symmetric random walk in $\Bbb{Z}$, so $S_0=0$ and for all $n\geq 1$, $S_n=X_0+…+X_n$ where $(X_i)_i$ is an i.i.d. sequence with $\Bbb{P}(X_1=\pm 1)=1/2$. Now let $\mathcal{F}_n=\sigma\left(S_j, 0\leq j\leq n\right)$. I want to show that the process $(S^2_n-n)_{n\geq 0}$ is $(\mathcal{F}_n)_{n\geq 0}$ adapted.
My problem is that I don't really know how to show this. I know that $(S_n)_n$ is $(\mathcal{F}_n)_n$ adapted by the definition of $(\mathcal{F}_n)_n$. And I know that I need to show by definition that for all $n$, $S^2_n-n$ is $\mathcal{F}_n$ measurable.
Can maybe someone give me a hint how to show this?
Best Answer
$\sigma (S_n^{2}-n)\subseteq \sigma (S_n) \subseteq \mathcal F_n $.