Consider the following versions:
$$dX_t=x_0+sgn(X_t)dW_t \tag1$$
$$dX_t=x_0+1_{(0,+\infty)}(X_t)dW_t \tag2$$
$$dX_t=x_0+1_{(-\infty,0]}(X_t)dW_t \tag3$$
SDE (1) is a classical example of SDE with no strong solution. SDE (2) has the following strong solution $X_t=x_0+W_{t\wedge \sigma}$, where $\sigma=\inf\{t\ge 0 : x_0+W_t\le 0\}$. However, SDE (3) is not similar to (2), e.g. if $x_0=0$, $X_t=0$ is not a solution of (3). Anyway, is there a solution of SDE (3) in some sense?
Edit:
It seems that the answer is negative. Set $x_0=0$ and take any twice cont. differentiable function $f$ with $f'(0)=0$ and $f''(0)=0$. Then if $X_t$ is a solution of (3), using Ito formula
\begin{align} f(X_t)1_{(0,\infty)}(X_t)&=\int_0^t f'(X_s)1_{(0,\infty)}(X_s)1_{(-\infty, 0]}(X_s)dW_s \\ &+\int_0^t f''(X_s)1_{(0,\infty)}(X_s)1_{(-\infty, 0]}(X_s)ds=0, \end{align}
which means that $X_t\le 0$ for all $t$ a.s. Plugging this into (3) implies that $X_t$ must be equal to $W_t$, which is not nonnegative for all $t$.
Does this make sense? Thank's!