Averaging Method for Neutral Stochastic Delay Differential Equations Driven by Fractional Brownian Motion

Peiguang Wang,Yan Xu

doi:10.1155/2020/5212690

Abstract

In this paper, we investigate the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H∈1/2,1. By using the linear operator theory and the pathwise approach, we show that the solutions of neutral stochastic delay differential equations converge to the solutions of the corresponding averaged stochastic delay differential equations. At last, an example is provided to illustrate the applications of the proposed results.

Highlights

Since Kolmogorov’s work began in 1940 [1], the fractional Brownian motion fBHðtÞ, t ≥ 0g with Hurst parameter H ∈ ð0, 1Þ has been studied by some authors [2, 3]
When H ≠ 1/2, the BHðtÞ neither is a semimartingale nor a Markov process. These properties mean that the analysis tools for the classical stochastic differential equation theory no longer work
Our aim is to study the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion

Summary

Introduction

Since Kolmogorov’s work began in 1940 [1], the fractional Brownian motion fBHðtÞ, t ≥ 0g with Hurst parameter H ∈ ð0, 1Þ has been studied by some authors [2, 3]. BHðtÞ is a zero mean Gaussian stochastic process with a covariance function determined by parameter H. It is selfsimilar and has the stationary increments and its increment process has long-range dependence when Hurst parameter 1/2 < H < 1, which makes BHðtÞ a suitable candidate to model many complex phenomena in finance and other practical problems. When H ≠ 1/2, the BHðtÞ neither is a semimartingale nor a Markov process. These properties mean that the analysis tools for the classical stochastic differential equation theory no longer work. The most obvious problem is how to define a proper notion of stochastic integral with respect to BHðtÞ. Three main integration techniques with regard to BHðtÞ have been researched, see, for example, [4,5,6,7,8] and the references therein

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