How to prove that $S_n^2 − Var(S_n )$ is a martingale

Question

I would be grateful for some help with the following exercise:

Let $(X_n ,n≥1)$ be a sequence of independent random variables with $E[X_i]=0$, and $Var(X_i)=σ_i^2<\infty, ∀i ∈\mathbb{N}$. Prove that $S_n^2 − Var(S_n)$ is a martingale, where $S_n:= \sum\limits_{i=1}^n X_i$

We have to show: $E[S_{n+1}^2-Var(S_{n+1})|\mathcal{F}_n]=S_n^2-Var(S_n)$.

I tried to do this by induction, but I'm already having trouble with the base case:

n=1: $E[S_2^2-Var(S_2)|\mathcal{F}_1]=E[X_1^2|\mathcal{F}_1]+2E[X_1|\mathcal{F}_1]E[X_2 \mathcal{F}_1]+E[X_2^2|\mathcal{F}_1]-E[E(X_1^2)+E(X_2^2)|\mathcal{F}_1]=...?$

I would be glad, if you could tell me, if I did something wrong so far, or, if not, how to continue.

Thanks for your help!

Answer 1

Here I will use $E_n[\cdot]$ for $E[\cdot | \mathcal{F}_n]$.Note that since the $X_i$ are independent, you have $E_n(\sum_1^N X_i)^2 = \sum_1^N E_n X_i^2$, all of the cross terms vanish since for $i < j$ we have by iterated conditioning $$E_nX_iX_j = E_nE_i X_iX_j = E_n X_i E_iX_j = E_n X_i E X_j = E_n X_i 0 = 0.$$ Use this and the fact variances add for independent random variables as well.