Open Access
2011 On the one-sided Tanaka equation with drift
Ioannis Karatzas, Albert Shiryaev, Mykhaylo Shkolnikov
Author Affiliations +
Electron. Commun. Probab. 16: 664-677 (2011). DOI: 10.1214/ECP.v16-1665


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".


Download Citation

Ioannis Karatzas. Albert Shiryaev. Mykhaylo Shkolnikov. "On the one-sided Tanaka equation with drift." Electron. Commun. Probab. 16 664 - 677, 2011.


Accepted: 31 October 2011; Published: 2011
First available in Project Euclid: 7 June 2016

zbMATH: 1243.60048
MathSciNet: MR2853104
Digital Object Identifier: 10.1214/ECP.v16-1665

Primary: 60H10
Secondary: 60J60 , 60J65

Keywords: comparison theorems for diffusions , skew Brownian motion , Sticky Brownian motion , Stochastic differential equation , strong existence , Strong uniqueness , Tanaka equation , weak existence , Weak uniqueness

Back to Top