Invariance and Monotonicity for Stochastic Delay Differential Equations

We study invariance and monotonicity properties of Kunita-type stochastic differential equations in ${\text{RR}}^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of ${\text{RR}}^d$. 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.