Conditional expectation inequality

Question

Let $F_1\subset F_2$ and $\mathbb{E}X^2 < \infty$, then

$\mathbb{E}|X-\mathbb{E}(X|F_2)|^2 \le \mathbb{E}|X-\mathbb{E}(X|F_1)|^2 $

Can anyone help me with this proof? I know basic properties of conditional expectation but can't see how to use them here. Thank you.

My attempt:

$Y:=X-\mathbb{E}(X|F_1)$

Then I will need to show that:

$\mathbb{E}(|Y-\mathbb{E}(Y|F_2)|^2|F_2) \le \mathbb{E}(Y^2|F_2)$

But i don't see how can I move from here. I tried to employ Jensen Inequality but it gave me nothing. Any clue?

Answer 1

Please

1 finish it

2 convince yourself if exchange the role of $F_1$ and $F_2$, the argument, mainly the bit at the bottom, does not work (not equal to 0).

Answer 2

First:

$\mathbb{E}[X-\mathbb{E}[X|F_2]|F_2]=\mathbb{E}[X|F_2]-\mathbb{E}[\mathbb{E}[X|F_2]|F_2]=\mathbb{E}[X|F_2]-\mathbb{E}[X|F_2]=0$

So I have.

$\mathbb{E}[X-\mathbb{E}[X|F_1]]^2=\mathbb{E}[X-\mathbb{E}[X|F_2]]^2+\mathbb{E}[\mathbb{E}[X|F_2]-\mathbb{E}[X|F_1]]^2\ge \mathbb{E}[X-\mathbb{E}[X|F_2]]^2$

Where the middle equality holds by the last bit of my answer, which tells us the cross term is 0

Is that correct? Thank you again!