Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and $$ dX_t=f_tdt+B_tdW_t $$ where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated by $W$.
Assume $x$ is a constant. One possible solution is that $f_t=B_t=0$ so that $X_t=x, \forall t$. Is it possible to have other solutions where $f$ or $B$ are not always 0?.