On the strong and weak solutions of one-dimensional stochastic equations with boundary conditions

We investigate the sufficient conditions for the strong and weak uniqueness of the solution of a stochastic equation with a simultaneous reflection at zero:
\begin{gather*} dx_s=\sigma(s,x_s)\,dw_s+b(s,x_s)\,ds+d\varphi_s, \\ d\varphi_s=\chi_{x_s=0}\,d\varphi_s,\qquad x_s\ge 0, \end{gather*}
and for the existence of a strong solution of this equation.