Pathwise uniqueness for a degenerate stochastic differential equation

We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation $$dX_t=|X_t|^{\alpha} dW_t,$$ where $W_t$ is a one-dimensional Brownian motion and $\alpha\in(0,1/2)$. Weak uniqueness does not hold for the solution to this equation. If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. We also consider a class of stochastic differential equations with reflection.