I'm studying filtration enlargements and I bumped into the following problem.
Assume there are two filtrations $\mathbb{F}=(\mathcal{F}_t)_{t \in T}$ and $\mathbb{H}=(\mathcal{H}_t)_{t \in T}$ on the same probability space $(\Omega, \mathcal{F},\mathbb{P})$ such that, for all $t \in T$,
$$ \mathcal{F}_t \subset \mathcal{H}_t $$
Consider the following statements: $$ (A) \qquad \mbox{every }\mathbb{F}-\mbox{martingale is a }\mathbb{H}-\mbox{semimartingale} $$
$$ (B) \qquad \mbox{every }\mathbb{F}-\mbox{local martingale is a }\mathbb{H}-\mbox{semimartingale} $$
Is it true that (A) implies (B)?
Some context:
I'm asking because I found that in literature both these conditions are known as hypothesis $H'$, in the sense that it's possible to find under the same name both the conditions
- Every $\mathbb{F}-$martingale is a $\mathbb{H}-$semimartingale
- Every $\mathbb{F}-$semimartingale is a $\mathbb{H}-$semimartingale
My gut guess is that 1 implies 2. To prove it, I noticed that it's enough to show that (A) implies (B) (as any $\mathbb{F}$-adapted process $F$ with finite variation is also a $\mathbb{H}$-adapted process with finite variation and the sum of a $\mathbb{H}-$semimartingale and $F$ it's again an $\mathbb{H}-$semimartingale).