Imperial College London
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Professor Xue-Mei Li

Faculty of Natural Sciences, Department of Mathematics

Chair in Probability and Stochastic Analysis
 
 
 
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Contact

 

+44 (0)20 7594 3709xue-mei.li Website CV

 
 
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Location

 

6M51Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Li:2011:10.1214/10-AOP585,
author = {Li, X-M and Scheutzow, M},
doi = {10.1214/10-AOP585},
journal = {Annals of Probability},
pages = {1407--1421},
title = {Lack of strong completeness for stochastic flows},
url = {http://dx.doi.org/10.1214/10-AOP585},
volume = {39},
year = {2011}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - It is well known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If, in addition, the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition x, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently of x, then the maximal flow is called strongly complete. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a two-dimensional SDE with coefficients which are even bounded (and smooth) and which is not strongly complete thus answering the question in the negative.
AU  - Li,X-M
AU  - Scheutzow,M
DO  - 10.1214/10-AOP585
EP  - 1421
PY  - 2011///
SN  - 0091-1798
SP  - 1407
TI  - Lack of strong completeness for stochastic flows
T2  - Annals of Probability
UR  - http://dx.doi.org/10.1214/10-AOP585
UR  - http://hdl.handle.net/10044/1/54121
VL  - 39
ER  -
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