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Lack of strong completeness for stochastic flows
- Source :
- Ann. Probab. 39, no. 4 (2011), 1407-1421
- Publication Year :
- 2011
- Publisher :
- Institute of Mathematical Statistics, 2011.
-
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 $x$, then the maximal flow is called {\em 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 2-dimensional SDE with coefficients which are even bounded (and smooth) and which is {\em not} strongly complete thus answering the question in the negative.
- Subjects :
- Statistics and Probability
Pure mathematics
37C10
Statistics & Probability
homogenization
weak completeness
stochastic differential equation
math.PR
Null set
Stochastic differential equation
SYSTEMS
Stochastic flow
Completeness (order theory)
FOS: Mathematics
Initial value problem
Almost surely
QA
Mathematics
Science & Technology
0104 Statistics
Probability (math.PR)
35B27
DIFFERENTIAL-EQUATIONS
Lipschitz continuity
EXISTENCE
Flow (mathematics)
Bounded function
Physical Sciences
MANIFOLDS
strong completeness
60H10
Statistics, Probability and Uncertainty
Mathematics - Probability
Subjects
Details
- ISSN :
- 00911798
- Volume :
- 39
- Database :
- OpenAIRE
- Journal :
- The Annals of Probability
- Accession number :
- edsair.doi.dedup.....3c8f916eec20a6ffcc9c8549eeff092c