Is it possible to find a strictly positive function $\sigma:\mathbb{R}\to\mathbb{R}$, such that a solution $X\_t$ to an SDE
$$dX\_t=-X\_tdt+\sigma(X\_t)\circ dB\_t,$$
with $X\_0$ being arbitrary, is a martingale? $B\_t$ denotes standard Brownian motion. I need to find an example of such a function if the answer is positive, or a proof if it is not.

I have tried the following. Writing the equation in Ito form, we have
$$dX\_t=\left(-X\_t+\frac{1}{2}\sigma(X\_t)\sigma'(X\_t)\right)dt+\sigma(X\_t)dB\_t.$$
So if I find a function $\sigma$ such that
$$X\_t=\frac{1}{2}\sigma(X\_t)\sigma'(X\_t)$$
then $X\_t$ is indeed a martingale. One such function is $\sigma(x)=\sqrt{2}x$, but it is not strictly positive. Any hints?