Let ${\{X_n}\}_{n \geq 0}$ and ${\{Y_n}\}_{n \geq 0}$ be two martingales. How do I show that $$\sum_{k=1}^n E[(X_k - X_{k-1})(Y_k - Y_{k-1})] = E[X_n Y_n] - E[X_0 Y_0]?$$
I tried expanding the terms and writing the first few terms of the sum out but nothing seems to be cancelling.
Any help is appreciated!
Since $\{X_n\}$ and $\{Y_n\}$ are martingales, for each $k\geq 1$ we have $$ \mathbb{E}[(X_k-X_{k-1})(Y_k-Y_{k-1})]=\mathbb{E}[X_kY_k]-\mathbb{E}[X_kY_{k-1}]-\mathbb{E}[X_{k-1}Y_k]+\mathbb{E}[X_{k-1}Y_{k-1}]$$ $$ =\mathbb{E}[X_kY_k]-\mathbb{E}[\mathbb{E}[X_kY_{k-1}\mid \mathcal{F}_{k-1}]]-\mathbb{E}[\mathbb{E}[X_{k-1}Y_{k}\mid \mathcal{F}_{k-1}]]+\mathbb{E}[X_{k-1}Y_{k-1}]$$ $$=\mathbb{E}[X_kY_k]-\mathbb{E}[Y_{k-1}\mathbb{E}[X_k\mid \mathcal{F}_{k-1}]]-\mathbb{E}[X_{k-1}\mathbb{E}[Y_{k}\mid \mathcal{F}_{k-1}]]+\mathbb{E}[X_{k-1}Y_{k-1}]$$ $$=\mathbb{E}[X_kY_k]-\mathbb{E}[X_{k-1}Y_{k-1}]$$ and hence the sum telescopes.