Lyapunov exponents of linear stochastic functional differential equations. Part II. Examples and case studies

• 成果类型: Article
• 署名作者: Mohammed, SEA; Scheutzow, MKR
• 署名单位: Southern Illinois University System; Southern Illinois University; Technical University of Berlin
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 1210-1240
• 关键词: stability

摘要

we give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semiflow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate <(lambda)over bar>(1) (sigma) of the trajectories expressed in terms of the noise variance sigma. Roughly speaking we show that for small sigma, <(lambda)over bar>(1) (sigma) behaves like -sigma(2)/2, while for large sigma, it grows like log sigma. In the regular case, it is shown that a discrete Oseledec spectrum exists, and upper estimates on the top exponent lambda(1) are provided. These estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases.