Let $\chi$ denotes an indicator variable and $W$ a Wiener process. What can we say about the solution of the following SDE:$$\begin{cases} dX_{t}=\left(2\chi_{\left\{ X_{t}>0\right\} }-1\right)\cos X_{t}dt+\cos X_{t}dW_{t}\\ X_{0}=0 \end{cases}$$
Is there any solution? Is there any kind of uniqueness in the solution if at least one solution exists? If there is uniqueness, then what kind of uniqueness is this (uniqueness in distribution, in trajectories...)?
I have got this exercise in our course, but the coefficient function: $\left(2\chi_{\left\{ x>0\right\} }-1\right)\cos x$ is not even continuous, so it can't be Lipschitz continuous (here $\chi$ is the indicator function). My answer would be that we can't say anything about the solution, however I generated some paths numerically, and I didn't feel any strangeness in it.