Stochastic differential equations for sticky Brownian motion

We study (i) the stochastic differential equation (SDE) systemfor Brownian motion X in sticky at 0, and (ii) the SDE systemfor reflecting Brownian motion X in sticky at 0, where X starts at x in the state space, is a given constant, is a local time of X at 0 and B is a standard Brownian motion. We prove that both systems (i) have a jointly unique weak solution and (ii) have no strong solution. The latter fact verifies Skorokhod's conjecture on sticky Brownian motion and provides alternative arguments to those given in the literature.