How to show $M_n = X_n^2-n$ is a martingale?

Question

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?

Answer 1

For $s\le n$, $$\mathbb{E}(M_{n+1}\mid \mathcal{F}_s)=\mathbb{E}((X_n+Y_{n+1})^2-(n+1)\mid \mathcal{F}_s)\\=X_n^2+2X_n\mathbb{E}(Y_{n+1})+\mathbb{E}(Y_{n+1}^2)-(n+1)\\=X_n^2-n=M_n$$ the steps have used the fact that $Y_n$ is an i.i.d. $\mathcal{N}(0,1)$ sequence. The possible mistake that you are making in your approach is the one rightly pointed out by @saz in the comments.

For part (B) $$\mathbb{E}M_n=\mathbb{E}X_n^2-n$$ Note that $X_n=\sum_{i=1}^n Y_i+X_0\implies \mathbb{E}X_n^2=n+X_0^2\implies \mathbb{E}M_n=X_0^2$

Answer 2

Let's look at $\mathbb{E}[X_{n+1}^2-(n+1)|F_n]$ (take $s=n$). We need show it equals to $X_n^2-n$. Since $X_{n+1}=X_n+ Y_{n+1}$ It should be $$ \mathbb{E}[X_{n}^2|F_n]+2\mathbb{E}[X_{n}Y_{n+1}|F_n]+\mathbb{E}[Y_{n+1}^2|F_n]-\mathbb{E}[n+1|F_n] $$, Since $X_{n}$ is measurable w.r.t. $F_n$, and $Y_{n+1}$ is independent of $F_n$ (for which we can "take out what is known, and by the assumption $Y_n$ is $\mathcal{N}(0,1)$ distributed, we get above equals

$$ X_{n}^2+2X_{n}\mathbb{E}[Y_{n+1}|F_n]+\mathbb{E}[Y_{n+1}^2]-(n+1), $$

$$ =X_{n}^2+2X_{n}\mathbb{E}[Y_{n+1}]+\mathbb{E}[Y_{n+1}^2]-(n+1). $$ Since $\mathbb{E}[Y_{n+1}]=0$, it's easy to see $\mathbb{E}[Y_{n+1}^2]=1$. Hence we reach $$ =X_{n}^2-n. $$