Proving Itô's Isometry

Question

Im currently trying to understand stochastic integration.

Reading through some lecture notes I encountered the following proof of Itô's Isometry for simple functions. Here $M_s$ is a $L^2$ integrable martingale starting at 0 and $$ H_t = \sum_{n = 0}^\infty K_n \mathbb{1}_{(t_{n},t_{n+1}]} $$ is a simple predictable process.

Then \begin{align} \mathbb{E}\left[\left( \int_0^\infty H_s dM_s \right)^2 \right] &\overset{1}{=} \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 (M_{t_{n+1}}-M_{t_n})^2 \right] \\ &\overset{2}{=}\mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 \mathbb{E}\left[(M_{t_{n+1}}-M_{t_n})^2 | \mathcal{F}_{t_n}\right] \right] \\ &\overset{3}{=} \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 \mathbb{E}\left[M_{t_{n+1}}^2-M_{t_n}^2 | \mathcal{F}_{t_n}\right] \right] \\ &\overset{4}{=} \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 \mathbb{E}\left[[M]_{t_{n+1}}-[M]_{t_n} | \mathcal{F}_{t_n}\right] \right] \\ &\overset{5}{=} \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2([M]_{t_{n+1}}-[M]_{t_n}) \right] \overset{6}{=} \mathbb{E}\left[ \int_0^\infty H_s^2 d[M]_s \right] \end{align}

I have trouble udnerstanding some parts of those equations. So lets go through each of them:

1. This is just the definition of the stochastic integral for simple functions.

2. This is the tower property for the conditional expectation.

3. I dont understand what is happending here. My calculations based on the comment by Bart: \begin{align} \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 \mathbb{E}\left[(M_{t_{n+1}}-M_{t_n})^2 | \mathcal{F}_{t_n}\right] \right] &= \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 \mathbb{E}\left[M_{t_{n+1}}^2-2M_{t_n}M_{t_{n+1}}-M_{t_n}^2 | \mathcal{F}_{t_n}\right] \right] \\ &= \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 \left( \mathbb{E}\left[M_{t_{n+1}}^2-M_{t_n}^2 | \mathcal{F}_{t_n}\right] -2\mathbb{E}\left[M_{t_n}M_{t_{n+1}}| \mathcal{F}_{t_n}\right] \right) \right] \\ &= \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 \left( \mathbb{E}\left[M_{t_{n+1}}^2-M_{t_n}^2 | \mathcal{F}_{t_n}\right] -2M_{t_n}^2 \right) \right] \\ &\overset{3}{=} \mathbb{E}\left[ \sum_{n =0}^\infty K_n^2 \mathbb{E}\left[M_{t_{n+1}}^2-M_{t_n}^2 | \mathcal{F}_{t_n}\right] \right] \\ \end{align} Here I still don't understand how to derive the last line.

4. I dont understand what is happending here. It is normally argued that $M_t^2-[M]_t$ is a martingale but i dont understand what this has to do with the proof.

5. Is this the tower property? I lack intuition why this is true.

6. $[M]_s$ is nondecreasing hence this integral is understood for example in the Riemann-Stieltjes sense.

Can you please help me understand this proof?

Answer 1

For 3:

There's a slight mistake when you expand out $(M_{t_{n+1}}-M_{t_n})^2$. It should be $M_{t_{n+1}}^2 - 2 M_{t_n} M_{t_{n+1}} {+} M_{t_n}^2$. The rest of what you have is good, but now you'll end up with $M_{t_n}^2 - 2 M_{t_n}^2 = -M_{t_n}^2$ in the last line.

For 4:

Let $N_t := M_t^2 - [M]_t$, which is a martingale as you mentioned. Given $t > s$, we have \begin{align*} \mathbb{E}[M_{t}^2 - M_{s}^2|\mathcal F_s] &= \mathbb{E}[M_t^2-[M]_t - (M_s^2 - [M]_s) + [M]_t - [M]_s | \mathcal F_s] \\ &= \mathbb{E}[N_t - N_s+ [M]_t - [M]_s | \mathcal F_s] \\ &= \mathbb{E}[[M]_t - [M]_s | \mathcal F_s]. \end{align*} Applying this with $t = t_{n+1}$ and $s = t_n$ gives the result.

For 5:

Yes, this is the tower property. We have \begin{align*} \mathbb{E}\left[\sum K_n^2 \mathbb{E}[[M]_{t_{n+1}} - [M]_{t_n} | \mathcal F_{t_n}] \right] &=\mathbb{E}\left[\sum \mathbb{E}[K_n^2([M]_{t_{n+1}} - [M]_{t_n} )| \mathcal F_{t_n}] \right] \\ &= \sum \mathbb{E}[\mathbb{E}[K_n^2([M]_{t_{n+1}} - [M]_{t_n} )| \mathcal F_{t_n}]]\\ &= \sum \mathbb{E}[K_n^2([M]_{t_{n+1}} - [M]_{t_n} )] \\ &= \mathbb{E}\left[ \sum K_n^2([M]_{t_{n+1}} - [M]_{t_n} )\right] \end{align*}