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August  2013, 18(6): 1533-1554. doi: 10.3934/dcdsb.2013.18.1533

## Invariance and monotonicity for stochastic delay differential equations

 1 Department of Mechanics and Mathematics, Kharkov National University, Kharkov, 61022, Ukraine 2 Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany

Received  January 2012 Revised  March 2012 Published  March 2013

We study invariance and monotonicity properties of Kunita-type sto-chastic differential equations in $\mathbb{R}^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\mathbb{R}^d$. Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (order-preserving) random dynamical system. Several applications are considered.
Citation: Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533
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