Lack of strong completeness for stochastic flows

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Title: Lack of strong completeness for stochastic flows
Authors: Li, X-M
Scheutzow, M
Item Type: Journal Article
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.
Issue Date: 5-Aug-2011
Date of Acceptance: 1-Aug-2011
ISSN: 0091-1798
Publisher: Institute of Mathematical Statistics
Start Page: 1407
End Page: 1421
Journal / Book Title: Annals of Probability
Volume: 39
Issue: 4
Copyright Statement: © Institute of Mathematical Statistics, 2011
Keywords: Science & Technology
Physical Sciences
Statistics & Probability
Stochastic flow
strong completeness
weak completeness
stochastic differential equation
0104 Statistics
Notes: mrclass: 60H10 mrnumber: 2857244
Publication Status: Published
Appears in Collections:Pure Mathematics
Faculty of Natural Sciences

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