One-dimensional stochastic differential equations with singular and degenerate coefficients

We show the existence of strong solutions and pathwise uniqueness for two types of one-dimensional stochastic differential equations. The first type allows singular drifts: X_t = X₀ + ∫₀ ^t a(X_t)dW_t + ∫_RL _t^w(X)μ(dw) for t ≥ 0, where W is a one-dimensional Brownian motion, a is a, function that is bounded between two positive constants, μ is a finite measure with |μ({w})| ≤ 1, and L^w is the local time at w for the semimartingale X. The second type is the equation dX_t = (X_t)ⁿdW_t + dL _t, where L is a continuous non-decreasing process that increases only when X is at 0, α ∈ (0, 1/2), and X_t ≥ 0 for all t. Although this second equation does not have a unique solution, it does have a unique solution if one restricts attention to those solutions that spend zero time at 0.