Let $(X_n)_{n \in \mathbb{N}}$ be independent random variables where $P(X_n = 2^{n}) = P(X_n = -2^{n}) = \frac{1}{2}$ and $M_n = \sum_{k=1}^n X_k$ for $n \in \mathbb{N}$.
Find the largest filtration, such that $(M_n)_{n \in \mathbb{N}}$ is a martingale.
I know the sigma-field $\sigma(M_1,...,M_n)$ is the smallest filtration that $M_n$ is adapted to, but how can I think of the largest?
In the end, all I need is that for $M_{n+1}$, $\mathcal{F_n}$ contains the informations for all previous $X_k$ and no information at tall on $X_{n+1}$
I am grateful for any tip and piece of advice