How to show $M_n = X_n^2-n$ is a martingale? Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk.
$X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$.
Then how can I show that:
A) $M_n = X_n^2-n$ is a martingale
Find the value of:
B) $\mathbb{E}M_{20}$
Does the answer lie in reformulation $X_n = X_{n+1} - Y_{n+1}$ ?
And then:
$$X_n = X_{n+1} - Y_{n+1} \implies X_n^2 = (X_{n+1})^2 - 2X_{n+1}Y_{n+1} + (Y_{n+1})^2$$
And then we proceed with
$$\mathbb{E}[(X_{n+1})^2 - 2X_{n+1}Y_{n+1} + (Y_{n+1})^2 -n| \mathcal{F}_s], s \leq n+1$$
$$\mathbb{E}[(X_{n+1})^2 | \mathcal{F}_s] - 2\mathbb{E}[X_{n+1}Y_{n+1}|\mathcal{F}_s] + \mathbb{E}[(Y_{n+1})^2|\mathcal{F}_s] - \mathbb{E}[n | \mathcal{F}_s]$$
Is this right so far? if not, where I have I made a mistake, and how should I proceed?
 A: $\mathbb{E} [X_{n+1}^2-(n+1) \space | \space \mathcal{F}_n \space] =\mathbb{E} [X_{n}^2+2X_n Y_{n+1}+Y_{n+1}^2-(n+1) \space | \space \mathcal{F}_n \space] $
This gives $X_n^2 + 2 X_n\mathbb{E} [Y_{n+1} \space | \space \mathcal{F}_n \space]+\mathbb{E} [Y_{n+1}^2 \space | \space \mathcal{F}_n \space]-n-1$.
Because ${\{Y_n\}}$ is iid standard normal, $\mathbb{E} [Y_{n+1}^2 \space | \space \mathcal{F}_n \space]$ and $\mathbb{E} [Y_{n+1} \space | \space \mathcal{F}_n \space]$ are 1 and 0 respectively.
Finally, we have $\mathbb{E} [M_{n+1} \space | \space \mathcal{F}_n \space] =X^2_n-n=M_n$
Therefore, we have $\{M_n\}$ being a martigale.
Second question,
$\mathbb{E} M_{20}=M_0=100 $
A: There is no mistake, your demo is right, but you must to go more deeply in the demo. If you want to demonstrate that your expression is a martingale you must express $X_{n}$ as a sum of the $Y_{n}$. After that, you would show the martingale property of your expression.
A: The another answer isn't quite rigorous because you can't write that
$\mathbb{E}[X_nY_{n+1}]=X_n\mathbb{E}[Y_{n+1}]$ directly.Firstly, you must to write that
$X_n=X_0+\sum_{k=0}^{n-1}Y_{k}$. As a result, you can use the independance between the $Y_k$ w.r.t to the filtration $\mathcal{F}_n$. 
