Let $M\_t$ be a continuous time real valued martingale, and $\mathcal F\_t$ its natural filtration.


Suppose that $\mathcal F\_t \setminus \mathcal F\_s$ is nonempty for all $t > s$.


Let $\mathcal G$ be a sigma algebra, and define the filtration $\mathcal H\_t := F\_t \vee \mathcal G$.


**Question:** Is it true that $M$ is a $\mathcal H\_t$ martingale if and only if $\mathcal G$ is independent of $\mathcal F\_t$ for all $t$?


*Remark: The if direction follows from a monotone class argument.*



Enlargement of filtration
I think that I have a counterexample. Let $(X,Y)$ be a Brownian motion in $\mathbb{R}^2$. Then $M = \int\_0^\cdot X\_s \mathrm{d}Y\_s$ is a martingale, in the natural filtration of $(X,Y)$, in its own filtration $(\mathcal{F}\_t)\_{t \ge 0}$ and also in $(\mathcal{F}\_t \vee \sigma(X))\_{t \ge 0}$. Yet, $X$ is not independent of $M$ since $\langle M \rangle = \int\_0^\cdot X\_s^2\mathrm{d}s$ is not deterministic.


Remark: in this example, $\mathcal{F}$ is not immersed in $\mathcal{H}$ since $X$ is no more a martingale in $\mathcal{H}$.


