Let $dX_t=u(t,w)dt+dB_t$, where $dB_t$ is a Brownian Motion, $u$ is bounded and measurable with respect to the filtration $F_t$ and $u$ be an ito process on $(\Omega, \{\mathcal{F}\},\mathcal{P})$. Find a martingale $M_t$ s.t. $M_0=1$ and $Y_t=X_tM_t$ is an $\mathcal{F}_t$-martingale.
It is easily derived that $X_t=X_0+\int^t_0 u(s,w)ds+B_t$.