Suppose that we have a random variable $Z$ and a process $X_t$ such that: $$dX_t=\mu(Z,X_t,t) dt+ dB_t$$ where $\mu(Z,X_t,t)$ is a smooth function that satisfies $E[\mu(Z,X_t,t)|X_t]=0 \ \ \forall t,X_t$ and $B_t$ is a Brownian motion.
Is $X_t$ generally a Brownian motion? I believe so, because it has quadratic variation and is a martingale. However, this seems strange to me as there is "more uncertainty" regarding the terminal value of $X_t$ than if we just had $dX_t=dB_t$. Can anyone provide me some insight here?
(Sorry for not including details on the underlying measure space. Maybe I need to to answer the question, but I'm not as familiar on those details as I would like).