How to show that a function $A_n$ is monotone increasing when $M_n^2 = N_n + A_n$, where $M_n, N_n$ are martingales

Question

Let $\{M_n,\mathcal{F}_n\}$ be a martingale with $\mathbb{E}(M_n^2) < \infty$ for all $n$. I would like to show that we can write $$ M_n^2 = N_n + A_n $$ where ${N_n, \mathcal{F}_n}$ is a martingale and $A_n$ is monotone increasing $A_n \geq A_{n-1}$. I am having a very hard time showing that it is monotone increasing. A hint that was given to me was to take $A_0 = 0$ and set $A_{n+1} = A_n + \mathbb{E}[(M_{n+1} - M_n)^2|\mathcal{F}_n].$ However, even with this I am failing to see how induction works. Would anyone have an idea? Thanks

Answer 1

Let $A_n$ as in the hint and $N_n=M_n^2-A_n$. Clearly $(A_n)$ is increasing and $M_n^2=N_n+A_n$. So all we have to show is that $N_n$ is a $\mathcal{F}_n$-martingale.

$E[N_{n+1}\mid\mathcal{F}_n]=E[M_{n+1}^2-A_{n+1}\mid\mathcal{F}_n]=E[M_{n+1}^2\mid\mathcal{F}_n]-E[A_n\mid\mathcal{F}_n]-E[E[(M_{n+1}-M_n)^2\mid\mathcal{F}_n]\mid\mathcal{F}_n]$

By induction, we see that $A_n$ is $\mathcal{F}_n$-measurable, so $E[A_n\mid\mathcal{F}_n]=A_n$.

We also have $E[E[(M_{n+1}-M_n)^2\mid\mathcal{F}_n]\mid\mathcal{F}_n]=E[(M_{n+1}-M_n)^2\mid\mathcal{F}_n]$.

This last part is $E[M_{n+1}^2\mid\mathcal{F}_n]-2E[M_nM_{n+1}\mid \mathcal{F}_n]+E[M_n^2\mid \mathcal{F}_n]$.

The middle term is $E[M_nM_{n+1}\mid \mathcal{F}_n]=M_n^2$ and the last term is also $M_n^2$ due to the properties of $M_n$ being a martingale.

When you put it all together you get $E[N_{n+1}\mid\mathcal{F}_n]=N_n$, which shows that $N_n$ is a $\mathcal{F}_n$-martingale.