Say I have a $\mathcal{F}_t$-adapted stochastic process $Y$ with Itô-representation $$Y_t=\int_0^t\alpha_sds+W_t$$ where both the Brownian motion and the integrable process $\alpha$ are $\{\sigma(Y_t)\}_{t}$-adapted. Moreover we know, that $\alpha$ is bounded by a constant $g\in \mathbb{R}$ and càdlàg, if that helps. Now I have a square-integrable martingale $M$ with respect to $\{\sigma(Y_t)\}_{t}$ and I want to use martingale representation theorem to get something like this $$M_t=M_0+\int_0^t\gamma_sdW_s.$$ Issue is I don't know if $M$ is $\{\sigma(W_t)\}_{t}$-adapted. Now obviously I can use Girsanov to show that $Y$ is a Brownian motion with respect to some other measure $\widetilde{P}$ on the same probability space, but I don't know if this enough. To be precise this question is motivated by this paper https://ieeexplore.ieee.org/document/179372 Theorem 1 step 3.
EDIT: If this was actually true, it would answer my question affirmatively -or would it?