Two martingales with respect to the same filtration

Question

Let $(X_n)_{n\ge1}$ and $(Y_n)_{n\ge1}$ be two martingales with respect to the same filtration $(\mathcal{F}_n)_{n\ge1}$. Suppose that $E[\lvert X_n\rvert]<\infty$ and $E[\lvert Y_n\rvert]<\infty$ for every $n\ge1$.

How do I prove the following assertions?

(1) $E[X_nY_n]-[X_1Y_1]=\sum\limits_{k=2}^n E[(X_k-X_{k-1})(Y_k-Y_{k-1})]$

(2) The random variables $X_1$, $X_k-X_{k-1}$, $k\ge2$, are pairwise orthogonal.

Thanks for your help!

Answer 1

Hint:

Let $\mathcal G$ denote a sigma-algebra and $\xi$ and $\eta$ some integrable random variables such that $\eta$ is measurable with respect to $\mathcal G$.

Then $E[\xi\eta\mid\mathcal G]=\zeta\eta$, where $\zeta=E[\xi\mid\mathcal G]$. In particular, $E[\xi\eta]=E[\zeta\eta]$.