I found following problems about properties of martingales and I would like to know is my approach correct for first problem and how to exactly solve second one.
Problem is following: Let $\{Y_n\}_{n\ge 0}$ be a martingale w.r.t. the filtration $\{F_n\}_{n\ge 0}$ and let $E(Y_n^2)<\infty$ $\forall$ n. Show that for arbitrary $i\le j\le k$, \begin{equation*} E((Y_k-Y_j)Y_i)=0 \end{equation*} and \begin{equation*} E[(Y_k-Y_j)^2|F_i]=E(Y_k^2|F_i)-E(Y_j^2|F_i). \end{equation*}
What I did with first one: \begin{align*} E((Y_k-Y_j)Y_i)&=E(Y_k\cdot Y_i-Y_j\cdot Y_i)\\ &=E(Y_k\cdot Y_i)-E(Y_j\cdot Y_i)\\ &=E[E(Y_k\cdot Y_i)|F_i]-E[E(Y_j\cdot Y_i)|F_i]\\ &=E[Y_i\cdot E(Y_k|F_i)]-E[Y_i\cdot E(Y_j|F_i)]\\ &=E[Y_i\cdot E(Y_k)]-E[Y_i\cdot E(Y_j)]\\ &=E[Y_i\cdot 0]-E[Y_i\cdot 0]= 0 \end{align*}
With second one I started like this: \begin{align*} E[(Y_k-Y_j)^2|F_i]&=E[(Y_k^2-2Y_kY_j-Y_j^2)|F_i]\\ &=E[(Y_k^2|F_i)]-2E[(Y_kY_j|F_i)]-E[(Y_j^2|F_i)]=\\ &=\dots \end{align*}
I have trouble showing following steps. I'm not sure that steps so far are correct ones. I hope someone helps me with those proofs.
Thanks!
In the first one the last two steps are wrong. You wrote $E(Y_k|F_i)=EY_k$. But $E(Y_k|F_i)=Y_i$ by martingale property. So you get $EY_i^{2}-EY_i^{2}=0$.
Now $E(Y_k-Y_j)^{2}|F_i)=E(Y_k^{2}|F_i)+E(Y_j^{2}|F_i)-2E(Y_kY_j|F_i)$. Write the last term as $-2 E(E(Y_kY_j|F_j|F_i))$. Then observe that $E(Y_kY_j|F_j)=Y_jE(Y_k|F_j)=Y_j^{2}$ by martingale property again. Now I hope you can finish the proof.