Comparison Theorem for Stochastic Differential Delay Equations with Jumps

TL;DR

This paper establishes a comparison theorem for stochastic differential delay equations with jumps, highlighting conditions under which the theorem holds or fails, especially concerning delay functions in diffusion and jump-diffusion terms.

Contribution

The paper provides new conditions for the validity of comparison theorems in stochastic delay equations with jumps, including counterexamples and specific cases.

Findings

Comparison theorem may not hold if diffusion term contains a delay.

Jump-diffusion coefficient with delay can still satisfy the theorem.

The theorem holds if jump-diffusion term is non-increasing in delay.

Abstract

In this paper we establish a comparison theorem for stochastic differential delay equations with jumps. An example is constructed to demonstrate that the comparison theorem need not hold whenever the diffusion term contains a delay function although the jump-diffusion coefficient could contain a delay function. Moreover, another example is established to show that the comparison theorem is not necessary to be true provided that the jump-diffusion term is non-increasing with respect to the delay variable.