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?.



Backward stochastic differential equation
The are surely many ways to do this. One classical example of this kind of process is the Brownian Bridge from $0$ to $x$, given by the SDE


$$
dX\_t = \frac{x - X\_t}{1-t}dt + dW\_t.
$$


This is solved by $X\_t = tx + (1-t)\int\_0^t{\frac{dW\_s}{1-s}}$. As $t$ approaches $1$, $X\_t$ approaches $x$ almost surely. 


