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

Authors: Peiguang Wang; Yan Xu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we investigate the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion with Hurst parameterH∈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.


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## Body

## 1. Introduction

Since Kolmogorov’s work began in 1940 [1], the fractional Brownian motion BHt,t≥0 with Hurst parameter H∈0,1 has been studied by some authors [2, 3]. BHt is a zero mean Gaussian stochastic process with a covariance function determined by parameter H. It is self-similar and has the stationary increments and its increment process has long-range dependence when Hurst parameter 1/2<H<1, which makes BHt a suitable candidate to model many complex phenomena in finance and other practical problems. When H≠1/2, the BHt 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 BHt. Three main integration techniques with regard to BHt have been researched, see, for example, [4–8] and the references therein.In recent years, there has been much interest in a stochastic averaging method, which provides a powerful tool to approximate the original dynamical systems under random fluctuations by a simpler system. The first work, which was introduced by Khasminskii [9], investigated a stochastic averaging method for stochastic differential equations with Gaussian random fluctuations. Since then, the stochastic averaging method has been developed for many types of stochastic differential equations, see, e.g., [10–15].Some phenomena of biological dynamics, engineering, and financial markets can be better understood when the effect of time delays is considered in the models, see, e.g., [16]. Hence, stochastic delay differential equations driven by fractional Brownian motion are proposed and have recently attracted great attention [17–19]. However, except [20] studied averaging method for stochastic delay differential equations of neutral type driven by G-Brownian motion, the averaging method for neutral stochastic delay differential equations is seldom considered. In this paper, our aim is to study the stochastic averaging method for neutral stochastic delay differential equations driven by fractional Brownian motion.


## 2. Model Description and Preliminaries

The aim of this section is to introduce the model and some preliminary lemmas.The following notations are needed in this paper.Let (Ω,F,Ftt≥0,P) be a complete probability space. BHt is a m-dimensional fractional Brownian motion with Hurst parameter H∈1/2,1 defined on the space (Ω,F,Ftt≥0,P). Let ∣∙∣ denote the Euclidean norm. Let ℂ=C−∞,0;ℝd denote the family of all continuous functions ξ:−∞,0⟶ℝd with the norm ξℂ=sup−∞<s≤0ξs. For a stochastic process ξ=ξt,t∈−∞,T,ξt∈ℂ defined by ξts=ξt+s,s∈−∞,0,t∈0,T. Throughout the paper, the symbol N stands for a generic constant, whose value can change from one line to another.In this paper, we will discuss the following neutral stochastic delay differential equations driven by fractional Brownian motion:(1)dxt+ctxt−τt=ft,xt,xt−τtdt+gt,xt,xt−τtdBHt,with initial condition x0=ξ∈C−∞,0;ℝd, Where c∈C−∞,T,0,1,0<T<∞. The function τ: 0,T⟶ℝ+ is the time delay. The mappings f=fid×1,fi:0,T×ℝd×ℝd⟶ℝ, g=gi,jd×m,gi,j:0,T×ℝd×ℝd⟶ℝ are Borel measurable.The following operator theory is useful to obtain our main results.Lemma 1 (see [21]).
Suppose thatD is a bounded linear operator on Banach space X, if D<1, then I−D has a bounded inverse operator I−D−1, and I−D−1≤1/1−D.Lemma 2 (see [21]).
Assume thatD is a bounded linear operator on Banach space X and has an inverse bounded operator, for arbitrary ΔD:X⟶X,ifΔD<1/D−1, then S=D+∆D has a bounded inverse, and S−1=∑j=0∞−1jD−1D−1ΔDj.
Let the Banach space(2)Q=x:x−∞,T∈C−∞,0;ℝd.
Φ:Q⟶Q be defined by
(3)Φxt=xt+ctxt−τt.Lemma 3.
LetcM=maxt∈−∞,Tct. If 0≤ct<1, then Φ has a bounded inverse Φ−1 on Q, and for all x∈Q(4)Φ−1xt=∑j≥0∏0≤i≤j−1−1jct−iτtxt−jτt,Φ−1x≤x/1−cM.Proof.
The proof processes are similar to Lemma 2.6 in [22], so we omit it here.
We note thatΦ and Φ−1 are both linear operators. Then, the system (1) can be changed by the inverse transformation of Φ into the following:
(5)dut=ft,Φ−1ut,Φ−1ut−τtdt+gt,Φ−1ut,Φ−1ut−τtdBHt,with initial condition u0=Φ−1x0. So, we can conclude that ut is a solution of system (5) if and only if Φ−1ut is a solution of system (1).
System (5) can be written in the integral form:
(6)ut=u0+∫0tfs,Φ−1us,Φ−1us−τsds+∫0tgs,Φ−1us,Φ−1us−τsdBHs,where ∫0tgs,xs,x,s−τsdBHs is a pathwise-defined integral which can be represented by fractional derivatives, see, e.g., Young et al. [4], Zähle [5], and Nualart and Răşcanu [7].LetH∈1/2,1,1−H<α<1/2, denoted by W−∞,Tα,∞, the space of measurable functions h:−∞,T⟶Rd with the norm
(7)ht=h∞,t+∫0th.+t−s−h.∞,st−s−α+1ds<∞,where h∞,t=sup−∞<s≤ths.Letα∈0,1/2 be fixed. Denoting W0α,1 the space of measurable functions h:0,T⟶ℝd such that
(8)hα,1≔∫0Thssαds+∫0T∫0shs−hys−yα+1dyds<∞.Forh∈W0α,1, defining the left-side fractional derivative D0+αhu by
(9)D0+αhu=1Γ1−αhuuα+α∫0uhu−hyu−y1+αdy10,Tu.Also denoted byWT1−α,∞ is the space of measurable functions h:0,T⟶ℝm such that
(10)h1−α,∞,T≔sup0≤s<t≤Tht−hst−s1−α+∫sthy−hsy−s2−αdy<∞.For anyh∈WT1−α,∞, setting ht−u≔hu−ht and defining the right-sided fractional derivative Dt−1−αht−u by
(11)Dt−1−αht−u=−11−αΓαhu−htt−u1−α+1−α∫uthu−hyy−u2−αdy10,tu.Forz∈WT1−α,∞, we have
(12)Λαz≔1Γ1−αsup0<s<t<TDt−1−αzt−s≤1Γ1−αΓαz1−α,∞,T<∞,where Γ· is the gamma function. Moreover, if h∈W0α,1 and z∈WT1−α,∞, one can define the integral ∫0thdz in the sense of Zähle [5] for all t∈0,T and it follows from Nualart and Răşcanu [7] that
(13)∫0thdz≤Λαzhα,1.For our purpose, we adopt the following hypotheses on the coefficients.(H1) The function gt,x,y is continuously differentiable in variables x and y and there exist some constants β,δ,0<β,δ≤1 such that the following properties hold for all x,y,x~,y~∈ℝd,t∈0,T:
(14)gt,x,y−gt,x~,y~≤Nx−x~+y−y~,∂xigt,x,y−∂xigt,x~,y+∂yigt,x,y−∂yigt,x,y~≤Nx−x~δ+y−y~δ,gt,x,y−gs,x,y+∂xigt,x,y−∂xigs,x,y+∂yigt,x,y−∂yigs,x,y≤Nt−sβ,∂xigt,x,y+∂yigt,x,y≤N,gt,x,y≤N1+x+y.(H2) The function ft,x,y is Lipschitz continuous and has linear growth in the variable x and y, uniformly in t∈0,T, that is, ft,x,y−ft,x~,y~≤Nx−x~+y−y~, ft,x,y≤N1+x+y.Letα0=min1/2,β,δ/1+δ. Under the hypotheses H1−H2 with β>1−H and δ>1/H−1, we can follow the work of Shevchenko [8] to testify that system (5) has a unique solution ut∈W−∞,Tα,∞, for all α∈1−H,α0. Hence, system (1) has a unique solution Φ−1ut=xt∈W−∞,Tα,∞. Furthermore, we can deduce g∈W0α,1 according to Nualart and Rᾰşcanu [7]. On the other hand, the trajectories of BH are almost surely locally β′-Hölder-continuous for all 0<β′<H see Zähle [5]. Then, BH belongs to the space WT1−α,∞. Hence, from the above discussion, we have the following estimate:
(15)∫0tgsdBHs≤ΛαBHgα,1,where ΛαBH has moments of all orders.Given thatϵ∈0,ϵ0 with ϵ0>0 is a fixed number, consider the following standard form of system (5)
(16)duϵt=ϵft,Φ−1uϵt,Φ−1uϵt−τtdt+ϵHgt,Φ−1uϵt,Φ−1uϵt−τtdBHt,with uϵ0=u0. Correspondingly, the standard form of system (1) is
(17)dxϵt+ctxϵt−τt=ϵft,xϵt,xϵt−τtdt+ϵHgt,xϵt,xϵt−τtdBHt,with xϵ0=x0.Letf¯x,y:ℝd×ℝd→ℝd and g¯x,y:Rd×Rd⟶Rd×m be measurable functions and satisfy the conditions in H1−H2 and the following inequalities:
(18)I11T~∫0T~fs,x,y−f~x,y2ds≤ψ1T~1+x2+y2,I21T~∫0T~gs,x,y−g~x,y2ds≤ψ2T~1+x2+y2,where T~∈0,T,ψiT~ are positive bounded functions with sup0≤T~≤TψiT~≤N,limT~→∞ψiT~=0,i=1,2.Letwϵt denote the solution to the following averaged system:
(19)dwϵt=ϵf¯Φ−1wϵt,Φ−1wϵt−τtdt+ϵHg¯Φ−1wϵt,Φ−1wϵt−τtdBHt,with wϵ0=u0. Then, yϵt≔Φ−1wϵt is the solution to the following averaged system:
(20)dyϵt=ϵf¯Φ−1yϵt,Φ−1yϵt−τtdt+ϵHg¯Φ−1yϵt,Φ−1yϵt−τtdBHt,with yϵ0=x0.Now, from the above hypotheses, we are in conditions to demonstrate the relationship betweenxϵt and yϵt.


## 3. Main Results and Proofs

The main goal of this section is to use the averaging principle to investigate the neutral stochastic delay differential equations driven by fractional Brownian motion. Theorem6 shows that the solution yϵt of the averaged system (20) converges to solution xϵt of the original system (17) in the sense of mean square.The following lemmas are crucial for our analysis and are proved in Shevchenko [8] and Nualart and Răşcanu [7].Lemma 4 (see [8]).
Assume that hypothesesH1−H2 hold. Then, there exists a constant G0>0, for t∈−∞,T such that
(21)EuϵT2≤G0.Lemma 5 (see [7]).
Assume that hypothesesH1−H2 hold. Then, there exists a constant G1>0, for s,t∈0,T, we have
(22)uϵt−uϵs≤G11+2Φ−1uϵTt−s1−α.
From Lemma5, we can prove similarly that wϵt−wϵs≤G11+2Φ−1wϵTt−s1−α.Theorem 6.
Assume that hypothesesH1−H2 and I1−I2 hold. For ϵ∈0,ϵ0 is a positive small parameter with ϵ0 a constant, θ>0, there exist constants ϵ1∈0,ϵ0,O>0 such that for any ϵ∈0,ϵ1,t∈0,Oϵ−λ,0<λ<1, we get
(23)sup0≤t≤Oϵ−λEuϵt−wϵt2≤θ,and then, we obtain
(24)sup0≤t≤Oϵ−λExϵt−yϵt2≤Φ−12θ.Proof.
According to systems (16) and (19), on account of the elementary inequality, we obtain
(25)Euϵt−wϵt2≤2ϵ2E∫0tfs,Φ−1uϵs,Φ−1uϵs−τs−f¯Φ−1wϵs,Φ−1wϵs−τsds2+2ϵ2HE∫0tgs,Φ−1uϵs,Φ−1uϵs−τs−g¯Φ−1wϵs,Φ−1wϵs−τsdBHs2≕J1t+J2t,where 0,t⊂0,b⊂0,T. On the one hand, for term J1t, we derive
(26)J1t=2ϵ2E∫0tfs,Φ−1uϵs,Φ−1uϵs−τs−f¯Φ−1wϵs,Φ−1wϵs−τsds2≤2bϵ2E∫0tfs,Φ−1uϵs,Φ−1uϵs−τs−f¯Φ−1wϵs,Φ−1wϵs−τs2ds.
On the other hand, applying (15) for term J2t, it follows that
(27)J2t≤4ϵ2HΛα2BHb1−2αE∫0tgs,Φ−1uϵs,Φ−1uϵs−τs−g¯Φ−1wϵs,Φ−1wϵs−τs2ds+4ϵ2HΛα2BHE∫0t∫0s1s−r1+αgs,Φ−1uϵs,Φ−1uϵs−τs−g¯Φ−1wϵs,Φ−1wϵs−τs−gr,Φ−1uϵr,Φ−1uϵr−τr+g¯Φ−1wϵr,Φ−1wϵr−τrdrds2≔J21t+J22t.
Furthermore, by (H1), Lemma 4 and Lemma 5 yields
(28)J22t≤4ϵ2HΛα2BH×E∫0t∫0sgs,Φ−1uϵs,Φ−1uϵs−τs−gr,Φ−1uϵr,Φ−1uϵr−τrs−r1+αdr+∫0sg¯Φ−1wϵs,Φ−1wϵs−τs−g¯Φ−1wϵr,Φ−1wϵr−τrs−r1+αdrds2≤Nϵ2HΛα2BHb21+β−α+Φ−12G121+4Φ−12G0b4−4α+G0b2−2α.
Hence, under hypothesesH1−H2, I1−I2 and Lemma 4, we get
(29)Euεt−wεt2≤2bε2E∫0tfs,Φ−1uεs,Φ−1uεs−τs−f¯Φ−1wεs,Φ−1wεs−τs2ds+4ε2HΛα2BHb1−2α×E∫0tgs,Φ−1uεs,Φ−1uεs−τs−g¯Φ−1wεs,Φ−1wεs−τs2ds+Nε2HΛα2BHb21+β−α+Φ−12G121+4Φ−12G0b4−4α+G0b2−2α≤Nε2b+ε2HΛα2BHb1−2αΦ−12E∫0tsup0≤σ≤suεσ−wεσ2ds+Nε2b2+ε2HΛα2BHb2−2α1+2Φ−12G0+Nε2HΛα2BHb21+β−α+Φ−12G121+4Φ−12G0b4−4α+G0b2−2α,
and so(30)sup0≤t≤bEuεt−wεt2≤Nε2b+ε2HΛα2BHb1−2αΦ−12E∫0bsup0≤σ≤suεσ−wεσ2ds+Nε2b2+ε2HΛα2BHb2−2α1+2Φ−12G0+Nε2HΛα2BHb21+β−α+Φ−12G121+4Φ−12G0b4−4α+G0b2−2α.
Then, selectλ∈0,1, O>0, such that for each t∈0,Oε−λ⊂0,T, we obtain
(31)sup0≤t≤Oε−λEuεt−wεt2≤Kε1−λ,
where(32)K=Nελ−11+2Φ−12G0ε2−2λO2+ε2H+2αλ−2λΛα2BHO2−2α+ε2HΛα2BHε2αλ−λ−βλO21+β−α+Φ−12G121+4Φ−12G0ε4αλ−4λO4−4α+G0ε2αλ−2λO2−2α×expNΦ−12ε2−2λO2+ε2H+2αλ−2λΛα2BHO2−2α.
Thus, for any given numberθ>0, we can choose ε1∈0,ε0 such that for each ε∈0,ε1 and t∈0,Oε−λ⊂0,T,
(33)sup0≤t≤Oε−λEuεt−wεt2≤θ,and hence,
(34)sup0≤t≤Oε−λExεt−yεt2=supt∈0,Oε−λEΦ−1uεt−Φ−1wεt2≤Φ−12θ.
The proof is therefore complete.Finally, as an application, we consider the following example.Example 1.
Consider the neutral stochastic delay differential equations driven by fractional Brownian motion:(35)dxεt+0.8xεt−τ=εft,xεt,xεt−τdt+εHgt,xεt,xεt−τdBHt,where the initial condition xε0=ξs,−∞<s≤0,Eξ2<∞, τ>0 is a constant, t∈0,T,
(36)ft,xεt,xεt−τ=2xεt+xεt−τcos2t,gt,xεt,xεt−τ=2,
BHt is a fractional Brownian motion. Then,
(37)f¯xεt,xεt−τ=1π∫0π2xεt+xεt−τcos2tdt=xεt+xεt−τ,g¯xεt,xεt−τ=2.LetΦxεt=xεt+ctxεt−τ. Thus, Φ−1xεt=∑j≥0∏0≤i≤j−1−1jct−iτxεt−jτ. Define a new averaged stochastic differential delay equation
(38)dyεt=εf¯Φ−1yεt,Φ−1yεt−τdt+εHg¯Φ−1yεt,Φ−1yεt−τdBHt,namely,
(39)dyεt=εΦ−1yεt+Φ−1yεt−τdt+2εHdBHt.Obviously, whent∈0,τ, the explicit solution of system (39) can be obtained as
(40)yεt=eεtξ+ε∫0te−εs∑i≥1−0.8iyεs−iτ+∑i≥0−0.8iyεs−i+1τds+2εH∫0te−εsdBHs.Whent∈τ,2τ, the explicit solution of system (2) is given by
(41)yεt=eεt−τyετ+ε∫τte−εs−τ∑i≥1−0.8iyεs−iτ+∑i≥0−0.8iyεs−i+1τds+2εH∫τte−εs−τdBHs.Repeating the same steps, we can deduce solutionyεt on the entire interval 0,T. It is easy to see that functions f,g,f¯,g¯ in systems (35) and (39) satisfy all the conditions in H1−H2 and I1−I2; therefore, Theorem 6 holds. That is,
(42)Exεt−yεt2≤Φ−12θ.



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