How do I show that this process is adapted with respect to a filtration?

Question

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?

Answer 1

$\sigma (S_n^{2}-n)\subseteq \sigma (S_n) \subseteq \mathcal F_n $.

Answer 2

To say that a sequence $(X_n)$ of random variables is adapted to $(\mathcal F_n)$ is to say that for each $n$, $X_n$ is $\mathcal F_n$ measureable. This latter phrase means that there is a (measurable) function $f_n:\Bbb Z^n\to\Bbb R$ such that $X_n=f_n(S_1,\ldots,S_n)$. In words: $X_n$ is to be a function of the reandom variables on the list of generators of $\mathcal F_n$, and the function may well depend on $n$.