On the averaging principle for stochastic delay differential equations with jumps

Wei Mao,Surong You,Xuerong Mao,Xiaoqian Wu

doi:10.1186/s13662-015-0411-0

Abstract

In this paper, we investigate the averaging principle for stochastic delay differential equations (SDDEs) and SDDEs with pure jumps. By the Itô formula, the Taylor formula, and the Burkholder-Davis-Gundy inequality, we show that the solution of the averaged SDDEs converges to that of the standard SDDEs in the sense of pth moment and also in probability. Finally, two examples are provided to illustrate the theory.

Highlights

The averaging principle for dynamical system is important in problems of mechanics, control and many other areas
Motivated by the above discussion, in this paper we study the averaging principle for a class of stochastic delay differential equations (SDDEs) with variable delays and jumps
In Section, we prove that the solution of the averaged SDDEs converges to that of the standard SDDEs in the sense of pth moment and in probability; in Section we consider the above results as regards the pure jump case

Summary

Introduction

The averaging principle for dynamical system is important in problems of mechanics, control and many other areas. What is more, associated with all the work mentioned above, we pay special attention to the fact that most authors just focus on the mean-square convergence of the solution of the averaged stochastic equations and that of the standard stochastic equations. They do not consider the general pth (p > ) moment convergence case. In Section , we prove that the solution of the averaged SDDEs converges to that of the standard SDDEs in the sense of pth moment and in probability; in Section we consider the above results as regards the pure jump case.

Averaging principle for Brownian motion case

Averaging principle for pure jump case

Conclusion