I am trying to solve the following question.
{$ξ_k$} is $F_n$-martingale difference (i.e. for every $n$, $E[ξ_n|F_{n-1}]=0 $ a.s. )
Also, for every $n$ , $E[ξ_n^2]<\infty$
Show that
$S_n=ξ_1+ξ_2+…+ξ_n$ is square integrable martingale, and
$M_n=S_n^2-V_n$ is $F_n$-martingale where $V_n = E[ξ_1^2|F_0]+E[ξ_2^2|F_1]+…+E[ξ_n^2|F_{n-1}]$
Please give me some advice.
That $S_n$ is square integrable should be clear; for the martingale property write $S_n=S_{n-1}+\xi_n$ and use linearity of conditional expectation.
Notice that $$M_n=(S_{n-1}+\xi_n)^2-\mathbb E[\xi_n^2\mid \mathcal F_{n-1}]-V_{n-1}=M_{n-1}+2\xi_nS_{n-1}+\xi_n^2-\mathbb E[\xi_n^2\mid \mathcal F_{n-1}]$$ and $M_k$ is $\mathcal F_k$-measurable.