Discrete martingales problem

Question

Hello I am encountering difficulties with the following problem, please help.

Let $\{ Y_n \}_{n \geq 0}$ be a martingale w.r.t. the filtration $\{ F_n \}_{n \geq 0}$, and let $\mathbb E(Y_n^2) < \infty$ for all $n$. Show that for arbitrary $i \leq j \leq k$

$$\mathbb E((Y_k − Y_j)Y_i) = 0$$ and $$\mathbb E[(Y_k − Y_j)^2|F_i] = \mathbb E(Y_k^2|F_i) − \mathbb E(Y_j^2|F_i)$$

Answer 1

[1] $E[Y_kY_i]=E[E[Y_kY_i|Y_i]]=E[Y_iE[Y_k|Y_i]]=E[Y^2_i]$

changing $k \rightarrow j$ and subtructing we have the first equality.

[2] $E[(Y_k-Y_j)^2|F_i]=E[Y_k^2|F_i]+E[Y_j^2|F_i]-2E[Y_kY_j|F_i]$

but now:

$E[Y_kY_j|F_i]=E[E[Y_kY_j|F_j]|F_i]$ $=E[Y_jE[Y_k|F_j]|F_i]=E[Y^2_j|F_i]$

This way we have the second equality.

Use has been made as suggested repeatedly of https://en.wikipedia.org/wiki/Law_of_total_expectation and of the Martingale property.