On the comparison theorem for multidimensional SDEs with jumps

TL;DR

This paper establishes a precise criterion for when the comparison theorem applies to multidimensional SDEs with jumps, extending the understanding of solution ordering in stochastic systems with discontinuities.

Contribution

It provides a necessary and sufficient condition for the comparison theorem to hold in multidimensional SDEs with jumps and matrix-valued SDEs with jumps, clarifying theoretical foundations.

Findings

Derived a necessary and sufficient condition for the comparison theorem

Extended comparison results to matrix-valued SDEs with jumps

Clarified theoretical criteria for solution ordering in jump processes

Abstract

In this note, we give a necessary and sufficient condition under which the comparison theorem holds for multidimensional stochastic differential equations (SDEs) with jumps and for matrix-valued SDEs with jumps.