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On the one-sided tanaka equation with drift. (English) Zbl 1243.60048
Summary: 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”.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes 60J65 Brownian motion
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