Martingale property for stochastic time changed Martingale processes

Question

I came across a Lemma and proof in Tankov & Cont, financial modelling with jumps. The lemma and proof can be seen here:

I have a hard time understanding the reason for the relation in the following part of the proof. Is it even correct?

Answer 1

That follows from optional stopping (as in the second sentence of the proof) because $(v_t)_{t\ge 0}$ is independent of $(M_t)_{t\ge 0}$.

On the other hand, the claim $$ E\{E[M(v_t)|\mathcal F_s\vee\mathcal F^v_t]|\mathcal F_s\}= E\{E[M(v_t)|\mathcal F^M_{v_s}\vee\mathcal F^v_t]|\mathcal F_s\} $$ is less obvious because $\mathcal F_s\vee\mathcal F_t^v \subset \mathcal F^M_{v_s}\vee \mathcal F_t^v$ and the inclusion may be strict; still it is true by the Tower Property, because $\mathcal F_s\subset \mathcal F_s\vee\mathcal F^v_t$.