Suppose we want to find a filtration, say $(\mathcal{F}_t)$, to ensure that $Z_t := B_{8t}-6C_t$ is a martingale, where $B$ and $C$ are two independent Brownian motions.
For this, we want to ensure that each $Z_t$ is $\mathcal{F}_t-$measurable. I am thinking that $\mathcal{F}_t = \sigma(B_{8s}, s \leq t) \cap \sigma(C_{s}, s \leq t)$ for all $t$ would do the trick, although I am not sure how I would confirm this. Moreover, I am not sure if scaling $C$ would have an impact here. I would appreciate any hints here if possible.