An Averaging Principle for Stochastic Differential Delay Equations with Fractional Brownian Motion

Yong Xu,Yongge Li,Bin Pei

doi:10.1155/2014/479195

Abstract

An averaging principle for a class of stochastic differential delay equations (SDDEs) driven by fractional Brownian motion (fBm) with Hurst parameter in(1/2,1)is considered, where stochastic integration is convolved as the path integrals. The solutions to the original SDDEs can be approximated by solutions to the corresponding averaged SDDEs in the sense of both convergence in mean square and in probability, respectively. Two examples are carried out to illustrate the proposed averaging principle.

Highlights

The averaging principle in stochastic dynamical systems is often used, and it is useful and effective for exploring stochastic differential equations (SDEs) in many different fields [1,2,3,4]
The standard stochastic differential delay equations (SDDEs) is defined as t
In the rest of the paper, we will consider the connections between the solution processes Zε(t) and Xε(t)

Summary

Introduction

The averaging principle in stochastic dynamical systems is often used, and it is useful and effective for exploring stochastic differential equations (SDEs) in many different fields [1,2,3,4]. SDEs with fBm have played an increasingly significant role in various fields of applications, such as hydrology, queueing theory, and mathematical finance [15,16,17,18]. Against this background, Xu et al presented an averaging principle for SDEs with fBm [19]. Stochastic differential delay equations (SDDEs) give a mathematical formulation for such kinds of systems. For this reason, SDDEs have attracted more and more attentions except the averaging principle for SDDEs driven by fBm [20,21,22]. The similar conclusion holds for SDDEs with fBm, where the stochastic differential or stochastic integral is of symmetric and backward types

Some Preliminaries

An Averaging Principle for SDDEs with fBm

Examples