On the one-sided Tanaka equation with drift
  We study questions of existence and uniqueness of weak and strong solutions for a one-sided Tanaka equation with constant drift \lambda. We observe a dichotomy in terms of the values of the drift parameter: for \lambda\leq 0, there exists a strong solution which is pathwise unique, thus also unique in distribution; whereas for \lambda>0, the equation has a unique in distribution weak solution, but no strong solution (and not even a weak solution that spends zero time at the origin). We also show that strength and pathwise uniqueness are restored to the equation via suitable "Brownian perturbations". 
Ioannis Karatzas, Albert N. Shiryaev, Mykhaylo Shkolnikov
math.PR