Pathwise uniqueness for a degenerate stochastic differential equation

Richard F. Bass; Krzysztof Burdzy; Zhen Qing Chen

doi:10.1214/009117907000000033

Pathwise uniqueness for a degenerate stochastic differential equation

Richard F. Bass, Krzysztof Burdzy, Zhen Qing Chen

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16 引用（Scopus）

摘要

We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation dX_t = |X _t|^α dW_t, where W_t 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.

源语言	英语
页（从-至）	2385-2418
页数	34
期刊	Annals of Probability
卷	35
期	6
DOI	https://doi.org/10.1214/009117907000000033
出版状态	已出版 - 2007
已对外发布	是

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Bass, R. F., Burdzy, K., & Chen, Z. Q. (2007). Pathwise uniqueness for a degenerate stochastic differential equation. Annals of Probability, 35(6), 2385-2418. https://doi.org/10.1214/009117907000000033

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AB - We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation dXt = |X t|α dWt, 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.

KW - Local times

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KW - Stochastic differential equations

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Pathwise uniqueness for a degenerate stochastic differential equation

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