The Annals of Probability

Pathwise uniqueness for a degenerate stochastic differential equation

Richard F. Bass, Krzysztof Burdzy, and Zhen-Qing Chen

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Abstract

We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation

dXt=|Xt|αdWt,

where Wt is a one-dimensional Brownian motion and α∈(0, 1/2). Weak uniqueness does not hold for the solution to this equation. If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. We also consider a class of stochastic differential equations with reflection.

Article information

Source
Ann. Probab., Volume 35, Number 6 (2007), 2385-2418.

Dates
First available in Project Euclid: 8 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1191860425

Digital Object Identifier
doi:10.1214/009117907000000033

Mathematical Reviews number (MathSciNet)
MR2353392

Zentralblatt MATH identifier
1139.60027

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

Keywords
Pathwise uniqueness weak uniqueness local times stochastic differential equations

Citation

Bass, Richard F.; Burdzy, Krzysztof; Chen, Zhen-Qing. Pathwise uniqueness for a degenerate stochastic differential equation. Ann. Probab. 35 (2007), no. 6, 2385--2418. doi:10.1214/009117907000000033. https://projecteuclid.org/euclid.aop/1191860425


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