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

Mohammed, Salah-Eldin A. ; Scheutzow, Michael K. R.

Ann. Probab., Tome 25 (1997) no. 4, p. 1210-1240 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1997-07-14
Classification:  Stochastic functional differential equations,  Lyapunov exponents,  Brownian motion,  exponential growth rate,  semimartingale,  Oseledec spectrum,  stochastic semi-flow,  stochastic delay equations,  Poisson process,  singular equations,  regular equations,  60H10,  60H20,  34K50,  93E15,  60H25