Electronic Communications in Probability

On the one-sided Tanaka equation with drift

Ioannis Karatzas, Albert Shiryaev, and Mykhaylo Shkolnikov

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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