On the one-sided tanaka equation with drift

Abstract

We study questions of existence and uniqueness of weak and strong solutions for a one-sided Tanaka equation with constant drift λ. We observe a dichotomy in terms of the values of the drift parameter: for λ ≤ 0, there exists a strong solution which is pathwise unique, thus also unique in distribution; whereas for λ > 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”. © 2011 Association for Symbolic Logic.