Expectation of multipliction of two functions

Question

Let $(X_n)$ is a martingale,such that $Y_n = X_n-X_{n-1}$ is a $L^2$ function for every $n \in \mathbb{N}$. Show that $E[Y_mY_n] =0 $ whenever m is not equal to n.

I tried it but I am not able to conclude anything. Where do we have to use that $Y_n$ is in $L^2$ ?

Thanks in advance

Answer 1

Let $n<m$. Then $E(Y_nY_m)=E(X_n-X_{n-1})(Y_m-Y_{m-1})$ Condition on $\mathcal F_{m-1}$ and note that $$E((X_n-X_{n-1})(Y_m-Y_{m-1})|\mathcal F_{m-1})$$ $$=(X_n-X_{n-1})E((Y_m-Y_{m-1})|\mathcal F_{m-1})=(X_n-X_{n-1})\times 0$$ because $E(Y_m| \mathcal F_{m-1})=Y_{m-1}$ by martingale property.