How to show that a sequence of random variables is non-anticipating?

Question

Let $\{M_n,\mathcal{F}_n\}$ be a martingale with $\mathbb{E}(M_n^2) < \infty$ for all $n$. Then, let us define the following relation: $$ M_n^2 = N_n + A_n $$ where ${N_n, \mathcal{F}_n}$ is a martingale, $A_n$ is monotone increasing $A_n \geq A_{n-1}$. Taking $A_0 = 0$ and setting $A_{n+1} = A_n + \mathbb{E}[(M_{n+1} - M_n)^2|\mathcal{F}_n]$, the first two are trivial to show. Now, I would like to show that $A_n$ is non-anticipating, that is, $A_n \in \mathcal{F}_{n-1}$ for all $n$. I am not really sure where to start here even. Does anyone have any ideas? thanks.

Answer 1

Observe that

$$A_n = M_n^2 - N_n$$

where $M_n^2, N_n \in m\mathscr F_n$.

It follows that $A_n \in m\mathscr F_n$

By definition of conditional expectation, $\mathbb{E}[(M_{n+1} - M_n)^2|\mathcal{F}_n] \in m\mathscr F_n$.

It follows that

$$A_n + \mathbb{E}[(M_{n+1} - M_n)^2|\mathcal{F}_n] (= A_{n+1}) \in m\mathscr F_n$$

QED

Btw, we don't say that $A_n \in \mathscr F_n$ because $A_n$ is a random variable and the elements of $\mathscr F_n$ are events.