I have difficulty understanding the proof of the below proposition from Karatzas and Shreve.
I am very confused with the martingale $M$ given. How is this a martingale? How do we get $E[M_k | \mathscr{F}_n] = sgn [A_n - E(A_n | \mathscr{F}_{n-1})]$ for $k>n$ and $E[sgn[A_n - E(A_n|\mathscr{F}_{n-1})] | \mathscr{F}_k] = E(M_n | \mathscr{F}_k)$ for $k =0,1,\dots, n$?
For instance, for $k>n$ we have $E[M_k | \mathscr{F}_n] = E[M_n | \mathscr{F}_n] = M_n$. For it to be a martingale, we should have $M_n = sgn[A_n - E(A_n | \mathscr{F}_{n-1})]$. But how does this hold?
And, we should also have $E[sgn[A_n - E(A_n | \mathscr{F}_{n-1}) | \mathscr{F}_{n-1}] = E(M_n | \mathscr{F}_{n-1})$. I can't see how these hold.
