Q:

Problem - Expected value and Martingales

I have a problem with the following exercise.   
Let $(X_n)$ be a limited martingale (there exists $M>0$ such that for each $n$ is valid $P(|X_n|<M)=1$). 
Show that for each $m>n$ is valid $E[(X_m-X_n)^2]=Var(X_m)-Var(X_n)$. 
I tried to solve the exercise in the following way:  
$E[(X_m-X_n)^2]=E[(X_m-X_n)]^2+Var(X_m-X_n) 
=(E[X_m]-E[X_n])^2+Var(X_m-X_n)=Var(X_m-X_n)
=Var(X_m)+Var(X_n)-2Cov(X_m,X_n)$  
I'm stuck here.
Thanks in advance for your help.

A:

Note that 
$$\mathbb{E}[(X_m-X_n)^2]=\mathbb{E}[\mathbb{E}[(X_m-X_n)^2\mid \mathcal{F}_n]]$$
If you expand out the square and use properties of conditional expectation and the fact that $\{X_n\}$ is a martingale, you should get
$$ \mathbb{E}[(X_m-X_n)^2]=\mathbb{E}[X_m^2]-\mathbb{E}[X_n^2]=\mathrm{var}(X_m)-\mathrm{var}(X_n)$$
with the last equality following from the fact that $\mathbb{E}[X_m]=\mathbb{E}[X_n]$.

