Order-Preservation for Multidimensional Stochastic Functional Differential Equations with Jump

TL;DR

This paper establishes new necessary and sufficient conditions for order-preservation in multidimensional stochastic functional differential equations with jumps, extending existing comparison theorems to more general settings.

Contribution

It provides the first comprehensive criteria for order-preservation in multidimensional stochastic functional differential equations with jumps, including non-Lipschitz coefficients.

Findings

Extended comparison theorems for multidimensional equations

New necessary conditions for order-preservation

Improved sufficiency conditions for stochastic differential equations

Abstract

Sufficient and necessary conditions are presented for the order-preservation of stochastic functional differential equations on $\R^d$ with non-Lipschitzian coefficients driven by the Brownian motion and Poisson processes. The sufficiency of the conditions extends and improves some known comparison theorems derived recently for one-dimesional equations and multidimensional equations without delay, and the necessity is new even in these special situations.