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Lack of strong completeness for stochastic flows
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Li, XM. and Scheutzow, Michael. (2011) Lack of strong completeness for stochastic flows. The Annals of Probability, Vol.39 (No.4). pp. 14071421. ISSN 00911798

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Official URL: http://dx.doi.org/10.1214/10AOP585
Abstract
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 twodimensional SDE with coefficients which are even bounded (and smooth) and which is not strongly complete thus answering the question in the negative.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics  
Divisions:  Faculty of Science > Mathematics  
Library of Congress Subject Headings (LCSH):  Stochastic processes, Stochastic differential equations  
Journal or Publication Title:  The Annals of Probability  
Publisher:  Institute of Mathematical Statistics  
ISSN:  00911798  
Official Date:  July 2011  
Dates: 


Volume:  Vol.39  
Number:  No.4  
Page Range:  pp. 14071421  
Identification Number:  10.1214/10AOP585  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access  
Funder:  Engineering and Physical Sciences Research Council (EPSRC)  
Grant number:  EP/E058124/1 (EPSRC)  
References:  [1] BAXENDALE, P. (1980/81).Wiener processes on manifolds of maps. Proc. Roy. Soc. Edinburgh 

URI:  http://wrap.warwick.ac.uk/id/eprint/38510 
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