Consider this extract from the above book:
My question: should the first highlight 12.1.b not be $$ E[{(M_n-M_0)}^2] = \sum_1^n E[{(M_k-M_{k-1})}^2] $$
It would seem more compatible with the second highlight 12.1.d and formula 12.1.a
Many thanks in advance !
Using the tower property and the martingale property, we find
$$\mathbb{E}(M_0 M_n) = \mathbb{E} \big( \mathbb{E}(M_0 M_n \mid \mathcal{F}_0) \big) = \mathbb{E}\big( M_0 \underbrace{\mathbb{E}(M_n \mid \mathcal{F}_0)}_{M_0} \big) = \mathbb{E}(M_0^2)$$
and therefore
$$\mathbb{E}((M_n-M_0)^2) = \mathbb{E}(M_n^2)-2 \mathbb{E}(M_n M_0) + \mathbb{E}(M_0^2) = \mathbb{E}(M_n^2) - \mathbb{E}(M_0^2).$$
This shows that
$$\mathbb{E}(M_n^2) = \mathbb{E}(M_0^2) + \sum_{k=1}^n \mathbb{E}((M_k-M_{k-1})^2)$$
is equivalent to saying
$$\mathbb{E}((M_n-M_0)^2) = \sum_{k=1}^n \mathbb{E}((M_k-M_{k-1})^2)$$