Doubly Perturbed Neutral Stochastic Functional Equations Driven by Fractional Brownian Motion

XU Liping and LI Zhi

School of Information and Mathematics,Yangtze University,Jingzhou434023,China.

Doubly Perturbed Neutral Stochastic Functional Equations Driven by Fractional Brownian Motion

XU Liping and LI Zhi?

School of Information and Mathematics,Yangtze University,Jingzhou434023,China.

.In this paper,we study a class of doubly perturbed neutral stochastic functional equations driven by fractional Brownian motion.Under some non-Lipschitz conditions,we will prove the existence and uniqueness of the solution to these equations by providing a semimartingale approximation of a fractional stochastic integration.

Fractional Brownian motion;doubly perturbed neutral functional equations;non-Lipschitz condition.

1 Introduction

As the limit process from a weak polymer model,the following doubly perturbed Brownian motion

was presented in Norris,Rogers and Williams[1],also as the scaling limit of some selfinteracting random walks[2],has attracted much interest from several directions,see Le Gall and Yor[3],Davis[4,5],Carmona,Petit and Yor[6],Perman and Werner[7], Chaumont and Doney[8,9],Werner[10],etc.

Following them,Doney and Zhang[11]have studied the following single perturbed stochastic functional equation:with the condition that σ,b are Lipschitz continuous functions.

Recently,under some non-Lipschitz conditions,Luo[12]obtained the existence and uniqueness of the solution to the following doubly stochastic functional equation

Hu and Ren[13]studied the existence and uniqueness of the solution to the following doubly perturbed neutral stochastic functional equation

and Liu and Yang[14]proved the existence and uniqueness of the solution to the following doubly perturbed neutral stochastic functional equation with Markovian switching and Poisson jumps

One solution for many SDEs is a semimartingale as well a Markov process.However, many objects in real world are not always such processes since they have long-range aftereffects.Since the work of Mandelbrot and Van Ness[15],there is an increasing interest instochasticmodelsbased onthefractional Brownianmotion.A fractional Brownianmotion(fBm)of Hurst parameter H∈(0,1)is a centered Gaussian process BH={BH(t),t≥0} with the covariance function

When H=1/2 the fBm becomes the standard Brownian motion,and the fBm BHneither is a semimartingale nor a Markov process if H/=1/2.However,the fBm BH,H＞1/2 is a long-memory process and presents an aggregation behavior.The long-memory property make fBm as a potential candidate to model noise in mathematical flnance(see[16]); in biology(see[17]);in communication networks(see,e.g.,[18]);the analysis of global temperature anomaly[19]electricity markets[20]etc.

The aim of this paper is to study the existence and uniqueness of solution of the following doubly perturbed neutral stochastic functional equation

The rest of this paper is organized as follows.In Section 2,we recall the deflnition of a stochastic integral with respect to LfBm from an approximate approach.Section 3 is devoted to giving the main results of the paper.An example is presented in Section 4 to illustrate the theory.

2 Preliminaries

In the last few decades,many differential ways have been introduced to constructed the fractional stochastic calculus(see,for instance,[21]).The main difflculties in studying fractional stochastic systems are that we cannot apply stochastic calculus developed by It?o since fBm is neither a Markov process nor a semimartingale,except for H=1/2. Recently,an approximate approach has been developed to avoid those difflculties(see, [22,23]and the references therein).Let us recall some fundamental results about this approach.

Let(?,F,{Ft}t≥0,P)be a flltered completeprobability spacesatisfying the usualcondition,which means that the flltration is a right continuous increasing family andF0contains all P-null sets.For every ε＞0 we deflne

For f is a deterministic function in L2[0,T],from the decomposition(2.1)we have

As ε→0,each term in the right-hand side of(2.2)converges in L2(?)to the same term where ε=0.Then,it is‘natural’to deflne(we can refer the reader to[24,25]for a general deflnition).

Deflnition 2.1.For f is a deterministic function in L2[0,T].The stochastic integral of f with respect to LfBm is deflned by

Lemma 2.1.For f is a deterministic function in L2[0,T].We can obtain the following estimate for the integral(2.3):

Proof.By applying H¨older’s inequality and Burkh¨older’s inequality we have

In order to obtain the existence and uniqueness of the solution to Eq.(1.6),we make the following assumptions:

(H1)There exists a function A:R+×R+→R+such that

for all t≥0,where A(t,u)is locally integrable in t for each flxed u＞0 and is continuous nondecreasing in u for each flxed t≥0 and Xt:R+→R,and for any constant C the differential equation

has a global solution.

(H2)There exists a function B:R+×R+→R+such that

for all t≥0 and Xt,Yt:R+→R,where B(t,u)is locally integrable in t for each flxed u＞0 and is continuous nondecreasing in u for each flxed t≥0 and for any constant C,if a non-negative function Ytsatisfles the following inequality:

for all t≥0,then Yt≡0.

(H3)There exists a positive constant K＞0 such that

for all t≥0 and Xt,Yt:R+→R.

(H4)K+|a|+|b|＜1.

3 Main results

The main results of our paper is the following theorem on the existence and uniqueness of the solution to the stochastic differential Eq.(1.6).

Proof.We introduce the following iteration procedure.Let

For each integer n＞0,we deflne Xnas follows:

In order to get the conclusion,we give three steps as follows:

Taking the maximal value on both sides of(3.3),by H¨older’s inequality,the Burkh¨older’s inequality,(H1)and Lemma 2.1,we use C to denotea genericconstant which may change from line to line in the rest of the work and get

where n=0,1,2,...,and Ytis a solution to the following equation:

Taking the maximal value on both sides of(3.5),by H¨older’s inequality,the Burkh¨older’s inequality and(H2),we can get

By(3.5)and the Fatou lemma,we can obtain

Owing to(H2),it is easy to get

Step 3:Let us showthat the solution to Eq.(1.6)is unique.Now suppose that Xt,t≥0, and Yt,t≥0,are two solutions to Eq.(1.6),we have

Taking the maximal value on both sides of(3.7),by H¨older’s inequality,the Burkh¨older’s inequality and(H2),we can get

By(H2),it is deduced that E(max0≤s≤t|Xs?Ys|2)≡0,then the solution to Eq.(1.6)is unique,and the proof is completed.

Remark 3.1.When g≡0,Eq.(1.6)reduces to

which was recently studied in Hu and Ren[13],that is to say,Theorem 5 of[13]has been generalized.

4 An example

In this section,an example is provided to illustrate the obtained theory.

Consider the following doubly perturbed stochastic functional equation driven by fractional Brownian motion of the Liouville form:

with the initial condition X0≡c≥0(constant),where a,b,α,β are constants and|a|+|b|＜1, g∈L2[0,T].In order to get a uniqueFt-adapted solution Xt,t≥0 to Eq.(4.1)by Theorem 3.1,let B(t,u)=φ(t)?(u)where ?:R+→R+is a continuous nondecreasing function such that

φ(t)is locally integrable.Here we present an example of such a function ?.Deflne

where ε＞0 is sufflciently small.

Acknowledgement

This research is partially supportedby theNNSFof China(No.11271093)and theScience Research Project of Hubei Provincial Department of Education(No.Q20141306).

We are grateful to anonymous referees for many helpful comments and valuable suggestions on this paper.

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?Corresponding author.Email addresses:lizhi csu@126.com(Z.Li),xlp211@126.com(L.P.Xu)

Received 14 May 2015;Accepted 24 August 2015

AMS Subject Classiflcations:60H15,60G15,60H05

Chinese Library Classiflcations:O211.63,O175.2