The expectation is zero

Question

I do not understand how they have $\mathbb{E}(A_{j-1}(M_j - M_{j-1})) = 0$ in the example. From the martingale property, we just conclude $\mathbb{E}(M_j - M_{j-1}) = 0$, don't we?

Thank you very much for any explanation.

Answer 1

The martingale property says that the conditional expectation is zero, i.e., $\mathbb{E}(M_j-M_{j-1}\mid {\cal F}_{j-1})=0$. This is much stronger than just $\mathbb{E}(M_j-M_{j-1})=0$.

Since the process $(A_n)$ is adapted, we know that $A_{j-1}$ is ${\cal F}_{j-1}$-measurable and so we can pull it out of the conditional expectation:

$$\mathbb{E}(A_{j-1}(M_j-M_{j-1}) \mid {\cal F}_{j-1})=A_{j-1}\,\mathbb{E}(M_j-M_{j-1}\mid {\cal F}_{j-1})=A_{j-1}\cdot0=0.\tag1$$ Taking expected values in (1) gives $\mathbb{E}(A_{j-1}(M_j-M_{j-1}))=0.$