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Lack of strong completeness for stochastic flows

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Li, X-M. (Xue-Mei), 1964- and Scheutzow, Michael. (2011) Lack of strong completeness for stochastic flows. The Annals of Probability, Vol.39 (No.4). pp. 1407-1421. ISSN 0091-1798

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Official URL: http://dx.doi.org/10.1214/10-AOP585

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 two-dimensional 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:	0091-1798
Date:	July 2011
Volume:	Vol.39
Number:	No.4
Page Range:	pp. 1407-1421
Identification Number:	10.1214/10-AOP585
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)
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URI:	http://wrap.warwick.ac.uk/id/eprint/38510