Is this solution of martingale problem correct?

Question

I'm reading this problem

and its solution

I think there is a typo in the highlighted part. It should be

$$\begin{array}{l} =\mathbb{E}\left[V_{n}^{2}\right]+\mathbb{E}\left[H_{n}^{2} \mathbb{E}\left[D_{n+1}^{2} |\mathcal{F}_{n}\right]\right] \\ =\mathbb{E}\left[V_{n}^{2}\right]+\sigma^{2} \mathbb{E}\left[H_{n}^{2}\right] \end{array}$$

Even so, I suspect that this is not true because we are only given $\mathbb{E}\left[D_{n}^{2}\right]=\sigma^{2}$, not $\mathbb{E}\left[D_{n}^{2} | \mathcal{F}_{n}\right]=\sigma^{2}$.

Could you please confirm if my observation is correct or I miss something? Thank you so much!

Answer 1

Here is @why's comment that answers my question. I post it here to remove this question from unanswered list. All credits are given to @why.

Ya I believe there is an error. What you should get is $\mathbb E V_n^2=\mathbb E H^2 \langle M \rangle_n$