LACK OF STRONG COMPLETENESS FOR STOCHASTIC FLOWS

• 成果类型: Article
• 署名作者: Li, Xue-Mei; Scheutzow, Michael
• 署名单位: University of Warwick; Technical University of Berlin
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP585
• 发表日期: 2011
• 页码: 1407-1421
• 关键词: differential-equations; EXISTENCE; MANIFOLDS; systems

摘要

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.