Lyapunov Exponents of Linear Stochastic Functional-Differential Equations. II. Examples and Case Studies

Authors

Salah-Eldin A. Mohammed, Southern Illinois University CarbondaleFollow
Michael K. R. Scheutzow, Technical University of Berlin

Comments

Published in Annals of Probability, 25(3), 1210-1240.

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 $\overlineλ₁$(σ) of the trajectories expressed in terms of the noise variance σ . Roughly speaking we show that for small σ, $\overlineλ₁$(σ) behaves like -σ² /2, while for large σ, it grows like logσ. In the regular case, it is shown that a discrete Oseledec spectrum exists, and upper estimates on the top exponent λ₁ are provided. These estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases.

Recommended Citation

Mohammed, Salah-Eldin A. and Scheutzow, Michael K. "Lyapunov Exponents of Linear Stochastic Functional-Differential Equations. II. Examples and Case Studies." (Jan 1997).