Stochastic differential equations driven by stable processes for which pathwise uniqueness fails

Richard F. Bass; Krzysztof Burdzy; Zhen Qing Chen

doi:10.1016/j.spa.2004.01.010

Stochastic differential equations driven by stable processes for which pathwise uniqueness fails

Richard F. Bass^*, Krzysztof Burdzy, Zhen Qing Chen

^*Corresponding author for this work

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

27 Citations (Scopus)

Abstract

Let Z_t be a one-dimensional symmetric stable process of order α with α∈(0,2) and consider the stochastic differential equation dX_t=φ(X_t-)dZ_t. For β<(1/α) ∧1, we show there exists a function φ that is bounded above and below by positive constants and which is Hölder continuous of order β but for which pathwise uniqueness of the stochastic differential equation does not hold. This result is sharp.

Original language	English
Pages (from-to)	1-15
Number of pages	15
Journal	Stochastic Processes and their Applications
Volume	111
Issue number	1
DOIs	https://doi.org/10.1016/j.spa.2004.01.010
Publication status	Published - May 2004
Externally published	Yes

Keywords

Crossing estimates
Pathwise uniqueness
Stable processes
Stochastic differential equations
Time change

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Bass, R. F., Burdzy, K., & Chen, Z. Q. (2004). Stochastic differential equations driven by stable processes for which pathwise uniqueness fails. Stochastic Processes and their Applications, 111(1), 1-15. https://doi.org/10.1016/j.spa.2004.01.010

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title = "Stochastic differential equations driven by stable processes for which pathwise uniqueness fails",

abstract = "Let Zt be a one-dimensional symmetric stable process of order α with α∈(0,2) and consider the stochastic differential equation dXt=φ(Xt-)dZt. For β<(1/α) ∧1, we show there exists a function φ that is bounded above and below by positive constants and which is H{\"o}lder continuous of order β but for which pathwise uniqueness of the stochastic differential equation does not hold. This result is sharp.",

keywords = "Crossing estimates, Pathwise uniqueness, Stable processes, Stochastic differential equations, Time change",

author = "Bass, {Richard F.} and Krzysztof Burdzy and Chen, {Zhen Qing}",

year = "2004",

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doi = "10.1016/j.spa.2004.01.010",

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T1 - Stochastic differential equations driven by stable processes for which pathwise uniqueness fails

AU - Bass, Richard F.

AU - Burdzy, Krzysztof

AU - Chen, Zhen Qing

PY - 2004/5

Y1 - 2004/5

N2 - Let Zt be a one-dimensional symmetric stable process of order α with α∈(0,2) and consider the stochastic differential equation dXt=φ(Xt-)dZt. For β<(1/α) ∧1, we show there exists a function φ that is bounded above and below by positive constants and which is Hölder continuous of order β but for which pathwise uniqueness of the stochastic differential equation does not hold. This result is sharp.

AB - Let Zt be a one-dimensional symmetric stable process of order α with α∈(0,2) and consider the stochastic differential equation dXt=φ(Xt-)dZt. For β<(1/α) ∧1, we show there exists a function φ that is bounded above and below by positive constants and which is Hölder continuous of order β but for which pathwise uniqueness of the stochastic differential equation does not hold. This result is sharp.

KW - Crossing estimates

KW - Pathwise uniqueness

KW - Stable processes

KW - Stochastic differential equations

KW - Time change

UR - http://www.scopus.com/inward/record.url?scp=1642501171&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2004.01.010

DO - 10.1016/j.spa.2004.01.010

M3 - Article

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SN - 0304-4149

VL - 111

SP - 1

EP - 15

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 1

ER -

Stochastic differential equations driven by stable processes for which pathwise uniqueness fails

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Keywords

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