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July 1997 Lyapunov exponents of linear stochastic functional-differential equations. II. Examples and case studies
Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
Ann. Probab. 25(3): 1210-1240 (July 1997). DOI: 10.1214/aop/1024404511

Abstract

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 semi-flow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate $\overline{\lambda}_1 (\sigma)$ of the trajectories expressed in terms of the noise variance ƒÐ . Roughly speaking we show that for small $\sigma$, $\overline{\lambda}_1 (\sigma)$ 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.

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Salah-Eldin A. Mohammed. Michael K. R. Scheutzow. "Lyapunov exponents of linear stochastic functional-differential equations. II. Examples and case studies." Ann. Probab. 25 (3) 1210 - 1240, July 1997. https://doi.org/10.1214/aop/1024404511

Information

Published: July 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0885.60043
MathSciNet: MR1457617
Digital Object Identifier: 10.1214/aop/1024404511

Subjects:
Primary: 34K50 , 60H10 , 60H20
Secondary: 60H25 , 93E15

Keywords: Brownian motion , exponential growth rate , Lyapunov exponents , Oseledec spectrum , Poisson process , regular equations , Semimartingale , singular equations , stochastic delay equations , Stochastic functional differential equations , stochastic semi-flow

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 3 • July 1997
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