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- W2093883713 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." @default.
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- W2093883713 date "2011-07-01" @default.
- W2093883713 modified "2023-10-17" @default.
- W2093883713 title "Lack of strong completeness for stochastic flows" @default.
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- W2093883713 doi "https://doi.org/10.1214/10-aop585" @default.
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