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Sufficient and necessary conditions are presented for the order-preservationof stochastic functional differential equations on $\R^d$ with non-Lipschitziancoefficients driven by the Brownian motion and Poisson processes. Thesufficiency of the conditions extends and improves some known comparisontheorems derived recently for one-dimesional equations and multidimensionalequations without delay, and the necessity is new even in these specialsituations.

Order-Preservation for Multidimensional Stochastic Functional Differential Equations with Jump