Electronic Communications in Probability

On the one-sided Tanaka equation with drift

Ioannis Karatzas, Albert Shiryaev, and Mykhaylo Shkolnikov

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Abstract

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".

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 58, 664-677.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465262014

Digital Object Identifier
doi:10.1214/ECP.v16-1665

Mathematical Reviews number (MathSciNet)
MR2853104

Zentralblatt MATH identifier
1243.60048

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

Keywords
Stochastic differential equation weak existence weak uniqueness strong existence strong uniqueness Tanaka equation skew Brownian motion sticky Brownian motion comparison theorems for diffusions

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Karatzas, Ioannis; Shiryaev, Albert; Shkolnikov, Mykhaylo. On the one-sided Tanaka equation with drift. Electron. Commun. Probab. 16 (2011), paper no. 58, 664--677. doi:10.1214/ECP.v16-1665. https://projecteuclid.org/euclid.ecp/1465262014


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