  In this paper we review and improve pathwise uniqueness results for some
types of one-dimensional stochastic differential equations (SDE) involving the
local time of the unknown process. The diffusion coefficient of the SDEs we
consider is allowed to vanish on a set of positive measure and is not assumed
to be smooth. As opposed to various existing results, our arguments are mainly
based on the comparison theorem for local time and the occupation time formula.
We apply our pathwise uniqueness results to derive strong existence and other
properties of solutions for SDEs with rough coefficients.
