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

Karatzas, Ioannis; Shiryaev, Albert N.; Shkolnikov, Mykhaylo

Electron. Commun. Probab. 16, 664-677 (2011).

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

Keywords:

stochastic differential equation; weak existence; weak uniqueness; strong existence; strong uniqueness; tanaka equation; skew Brownian motion; sticky Brownian motion; comparison theorems for diffusions