Electronic Journal of Probability

Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations

Gerald Kager and Michael Scheutzow

Full-text: Open access

Abstract

Let $Z$ be an $R^m$-valued semimartingale with stationary increments which is realized as a helix over a filtered metric dynamical system $S$. Consider a stochastic differential equation with Lipschitz coefficients which is driven by $Z$. We show that its solution semiflow $\phi$ has a version for which $\varphi(t,\omega)=\phi(0,t,\omega)$ is a cocycle and therefore ($S$,$\varphi$) is a random dynamical system. Our results generalize previous results which required $Z$ to be continuous. We also address the case of local Lipschitz coefficients with possible blow-up in finite time. Our abstract perfection theorems are designed to cover also potential applications to infinite dimensional equations.

Article information

Source
Electron. J. Probab., Volume 2 (1997), paper no. 8, 17 pp.

Dates
Accepted: 2 December 1997
First available in Project Euclid: 26 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1453839984

Digital Object Identifier
doi:10.1214/EJP.v2-22

Mathematical Reviews number (MathSciNet)
MR1485117

Zentralblatt MATH identifier
0888.60044

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 28D10: One-parameter continuous families of measure-preserving transformations 34C35

Keywords
stochastic differential equation random dynamical system cocycle perfection

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Kager, Gerald; Scheutzow, Michael. Generation of One-Sided Random Dynamical Systems by Stochastic Differential Equations. Electron. J. Probab. 2 (1997), paper no. 8, 17 pp. doi:10.1214/EJP.v2-22. https://projecteuclid.org/euclid.ejp/1453839984


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