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

University of Connecticut
University of Washington

Research output: Contribution to journal › Article › peer-review

16 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	2385-2418
Number of pages	34
Journal	Annals of Probability
Volume	35
Issue number	6
DOIs	https://doi.org/10.1214/009117907000000033
Publication status	Published - 2007
Externally published	Yes

Keywords

Local times
Pathwise uniqueness
Stochastic differential equations
Weak uniqueness

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10.1214/009117907000000033

Cite this

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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 = |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.",

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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

KW - Pathwise uniqueness

KW - Stochastic differential equations

KW - Weak uniqueness

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JO - Annals of Probability

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Pathwise uniqueness for a degenerate stochastic differential equation

Abstract

Keywords

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