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

Gerald Kager (Technische Universität Berlin)
Michael Scheutzow (Technische Universität Berlin)

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.

 


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Pages: 1-17

Publication Date: December 2, 1997

DOI: 10.1214/EJP.v2-22

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