I know that if $H_t$ is a bounded and predictable/previsible (sub)martingale, and $X_t$ is a (sub)martingale, then $(H \cdot X)$ is also a (sub)martingale, and this is quite easy to show by using the linearity and stability properties of condiotional expectations.
Now, my question is, does it hold that if again $H_t$ is a bounded and predictable martingale, and $(H \cdot X)$ is martingale, then $X$ is martingale?
(I'm presuming time is discrete for this question.)
Let $(X_t)$ be adapted (to $(\mathcal F_t)$) with each $X_t$ integrable.
Suppose that $H\cdot X$ is a martingale for all $H$ of the form $H_t=1_A\cdot 1_{\{t\ge n+1\}}$, where $n\ge 0$ and the event $A$ is $\mathcal F_n$ measurable. For such an $H$ the martingale property of $H\cdot X$ implies that $E[(H\cdot X)_t]=0$ for each $t\ge 1$. This implies that $E[X_t-X_n; A]=0$ for each $t>n$, which in turn implies that $X$ is a martingale.