# Averaging Principle for Caputo Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion with Delays

Authors: Pengju Duan; Hao Li; Jie Li; Pei Zhang
Journal: Complexity
 (2021)

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## Abstract

In this article, we investigate a class of Caputo fractional stochastic differential equations driven by fractional Brownian motion with delays. Under some novel assumptions, the averaging principle of the system is obtained. Finally, we give an example to show that the solution of Caputo fractional stochastic differential equations driven by fractional Brownian motion with delays converges to the corresponding averaged stochastic differential equation.


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## Body

## 1. Introduction

Fractional differential equations have been developing as an active area on medicine, electrical engineering, biochemistry, and mechanical systems [1–5]. Because the systems are often subjected to noisy fluctuations, it is important to consider randomness into models. Since the fractional Brownian motion (fBm for short) owns many excellent properties, for example, long-range dependence and self-similar, it is usually used to describe the uncertainty. Since then, stochastic calculus with respect to fBm has been paid much attention in the stochastic analysis field, and many interesting works have obtained both qualitative and quantitative properties of stochastic differential equations (SDEs for short) driven by fBm [6–8]. Furthermore, the applications of fractional stochastic differential equations (FSDEs for short) driven by fBm have been widely applied in mathematical quantum, physics, and biology [9–11].For the deterministic systems, many varieties of methods are proposed for average systems, such as gradient-based and least squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle, two-stage least squares-based iterative estimation algorithm for CARARMA system modeling, decomposition-based fast least squares algorithm for output error systems, and gradient-based and least squares-based iterative estimation algorithms for multi-input multi-output systems [12–14]. Compared to the deterministic systems, due to influence of stochastic factors, the exact solution of FSDEs driven by fBm is difficult to realize, and the abovementioned methods do not work. Because the averaging principle shows that the complex original systems can be ignored and one can only concentrate on the average systems instead, it is usually taken as an effective tool to reduce the amount of calculation of the original systems. Khasminskii [15] first started with the averaging method to approximate the complex system with a simpler system. In recent years, the averaging method has been developed in many ways [16–18]. For example, Pei et al. [19] investigated stochastic averaging for stochastic differential equations driven by fBm and Brownian motion. Recently, Luo et al. [20] discussed the averaging principle for FSDEs of Itô-Doob with delays driven by Brownian motion. Xu et al. [21] presented the averaging principle for stochastic differential equations with Caputo fractional derivative.Inspired by the above works, we will discuss averaging principle of a new kind of SDEs with Caputo derivative driven by fBm and Brownian motion, which is a general case of [19, 20]. On the other hand, because Lipschitz conditions restrict the application, we will adopt weakened Lipschitz conditions to obtain the result. Moreover, in order to overcome the influence of Caputo derivative and fBm, we introduce a new averaging method to realize the stochastic averaging principle.In this article, we will deal with averaging principle of the following Caputo FSDEs driven by fBm and Brownian motion with delays:(1)DtαXt=ft,Xt,Xt+gt,Xt,XtdWtdt+σt,Xt,XtdBtHdt,t∈0,T,Xt=φt,−τ≤t≤0,where Dtα is the Caputo fractional derivative, α∈1/2,1. f:0,T×U×B⟶U,g:0,T×U×B⟶ℒ20V,U,σ:0,T×U×B⟶ℒ20V,U. BtH is a V-valued Q-cylindrical fBm with the Hurst parameter H∈1/2,1, Wt is a standard Wiener process on a real and separable Hilbert space V independent of BtH, and Xt=Xt+θ,θ∈−τ,0 is the B value stochastic process. The initial value φ=φθ:−τ≤θ≤0 is an ℱ0-measurable B-valued random variable independent of fBm BtH and Wiener process Wt with finite second moment.The rest part is arranged as follows. Section2 is devoted to some preliminary results and assumptions. In Section 3, the averaging principle is presented. An example is provided to show the result in Section 4.


## 2. Preliminary

LetΩ,ℱ,ℙ be a completed probability space. For any t∈0,T, ℱt denotes the σ field generated by BsH,Ws,s∈0,t, and all ℙ null sets. A one-dimensional fractional Brownian motion with Hurst parameter H∈0,1 is a centered Gaussian process βH=βHt with the covariance function:(2)Rt,s=EβHtβHs=12t2H+s2H−t−s2H.WhenH>1/2, βHt has the following representation:(3)βHt=∫0tKt,sdβs,where βs is a standard Brownian motion, and the kernel Kt,s is given by(4)Kt,s=cHs1/2−H∫stu−sH−3/2uH−1/2du,t≥s,where cH is a nonnegative constant with respect to H.For functionφ∈L20,T, the fractional Wiener integral of φ with respect to βH is defined by(5)∫0TφsdβHs=∫0TKH∗φsdβs,where KH∗φs=∫sTφt∂Kt,s/∂tdt.V,⋅V and U,⋅U are the two real separable Hilbert spaces with their vectors norms. Let ℒV,U denote the collection of all linear-bounded operators from V to U equipped with the norm ⋅. For the sake of convenience, we shall use the same notation ⋅ to denote the norms in V,U,ℒV,U. Q∈ℒV,U is an operator defined by Qen=λnen with finite trace trQ=∑n=1+∞λn<+∞, where λnn=1,2,… are nonnegative real numbers, and enn=1,2,… denotes a complete orthonormal basis in V. We define the infinite dimensional fBm on V with covariance Q as(6)BtH=∑n=1∞βnHtQ1/2en=∑n=1∞λnenβnHt,where βnHt are the real, independent fBms. The process BtH called V-valued Q-fBm, starts from 0, has zero mean and covariance:(7)EBtH,xBsH,y=Rt,sQx,y,for allx,y∈Vandt,s∈0,T.In the following parts, we shall introduce Wiener integral with respect to theQ-fBm BtH. Let ℒ20=ℒ20V,U denote the space of all Q-Hilbert–Schmidt operators ψ:V⟶U equipped with the norm(8)ψℒ20=∑n=1∞λnψen2<∞,and the inner product φ,ψℒ20=∑n=1∞φen,ψen for φ,ψ∈ℒ20.Now, we give the definition of the fractional Wiener integral of the functionψ:0,T⟶ℒ20 with respect to Q-fBm as follows:(9)∫0tψsdBsH=∑n=1∞∫0tψsQ1/2endβnHs=∑n=1∞∫0tKH∗ψQ1/2ensdβns,where βn is the standard Brownian motion with respect to βnH.We introduceB−τ,0,L2Ω,U (B for simply) denotes the family of all ℱ0-measurable bounded continuous functions ξ:−τ,0⟶L2Ω,U endowed with the norm ξt2=sup−τ≤θ≤tEξθ2. It is important of the following lemma to prove our main results, which is appeared in [22].Lemma 1.
Ifψ:0,T⟶ℒ20V,U satisfies(10)∫0Tψtℒ202dt<∞,then, for any 0≤s≤t≤T,(11)E∫stσrdBrH2≤CHt−s2H−1∫stσrℒ202dr.Now, we recall some notations and preliminary results about fractional calculus and some special functions.Definition 1.
For anyα∈0,1 and function f:0,T⟶U, the Riemann–Liouville fractional integral operator of order α is defined(12)Iαft=1Γα∫0tt−sα−1fsds,0≤t≤T,where Γα=∫0∞rα−1e−rdr.Definition 2.
The Caputo fractional derivative with orderα of function ft∈Cn0,T;U is defined as(13)Dtαft=1Γn−α∫0tfnst−sα+1−nds,n−1<α<n,dndtnft,α=n.In order to study the averaging principle of the system (1), we impose the following assumptions on the coefficient functions.Assumption 1.
For eachxi∈U,yi∈B,i=1,2, there exists a nonnegative function λt, such that(14)ft,x1,y1−ft,x2,y22∨gt,x1,y1−gt,x2,y2ℒ202∨σt,x1,y1−σt,x2,y2ℒ202≤λtx1−x22+y1−y22,where sup0≤t≤Tλt<+∞.Assumption 2.
For eachT1∈0,T,x∈U and y∈B, there exist positive bounded functions λit,i=1,2 and measurable functions f¯:U×R⟶U,g¯σ¯:U×R⟶ℒ20V,U, such that(15)1T1∫0T1t−s2α−2fs,x,y−f¯x,y2ds≤λ1T1x2+y2,1T1∫0T1t−s2α−2gs,x,y−g¯x,yℒ202ds≤λ2T1x2+y2,1T1∫0T1t−s2α−2σs,x,y−σ¯x,yℒ202ds≤λ3T1x2+y2.Remark 1.
In Assumption1, if we let λt=C, then it becomes the Lipschitz condition.Remark 2.
If we letα=1, then (16) is the common stochastic delay differential equation driven by fBm, and the Assumption 2 turns to classical averaging principle.Definition 3.
AU value stochastic process Xt−τ≤t≤T is called a mild solution of (16) if Xt satisfies the following:(1)
Xt is continuous and ℱt-measurable, Xt is a B value stochastic process;(2)
Fort∈−τ,0, Xt=φt(3)
For each0≤t≤T, Xt satisfies the following integral equation:(16)Xt=X0+1Γα∫0tt−sα−1fs,Xs,Xsds+1Γα∫0tt−sα−1gs,Xs,XsdWs+1Γα∫0tt−sα−1σs,Xs,XsdBsH,Xt=ξ,−τ≤t≤0.Remark 3.
If we letσt,⋅,⋅=0, then (16) becomes stochastic differential equations with Caputo fractional derivative in [21]. Moreover, the averaging principle in [21] is the special case of this study.


## 3. Main Results

In this section, combining the existence and uniqueness results in the second part, we investigate the averaging principle for the Caputo FSDEs. Let us consider the standard form of (16):(17)Xɛt=X0+ɛΓα∫0tt−sα−1fs,Xɛs,Xsɛds+ɛΓα∫0tt−sα−1gs,Xɛs,XsɛdWs+ɛHΓα∫0tt−sα−1σs,Xɛs,XsɛdBsH,where ɛ∈0,ɛ0 is a positive small parameter with ɛ0 being a fixed number.The following step is to introduce the original solutionXɛt converges, as ɛ tends to zero, to the solution Yɛt of the averaged system:(18)Yɛt=X0+ɛΓα∫0tt−sα−1f¯Yɛs,Ysɛds+ɛΓα∫0tt−sα−1g¯Yɛs,YsɛdWs+ɛHΓα∫0tt−sα−1σ¯Yɛs,YsɛdBsH.Theorem 1.
Suppose that Assumptions1 and 2 hold. Then, for a given arbitrary small δ>0, there exist constants L>0,ɛ1∈0,ɛ0 and β∈0,1, such that for all ɛ∈0,ɛ1,(19)supt∈−τ,Lɛ−βEXɛt−Yɛt2≤δ.Proof.
Based on the standard forms of (17) and (18), it deduces(20)Xɛt−Yɛt=ɛΓα∫0tt−sα−1fs,Xɛs,Xsɛ−f¯Yɛs,Ysɛds+ɛΓα∫0tt−sα−1gs,Xɛs,Xsɛ−g¯Yɛs,YsɛdWs+ɛHΓα∫0tt−sα−1σs,Xɛs,Xsɛ−σ¯Yɛs,YsɛdBsH.
Fort∈0,T, by the elementary inequality, we have(21)EXɛt−Yɛt2≤3ɛ2Γα2E∫0tt−sα−1fs,Xɛs,Xsɛ−f¯Yɛs,Ysɛds2+3ϵΓα2ɛE∫0tt−sα−1gs,Xɛs,Xsɛ−g¯Yɛs,YsɛdWs2+3ɛ2HΓα2E∫0tt−sα−1σs,Xɛs,Xsɛ−σ¯Yɛs,YsɛdBsH2≕I1+I2+I3.
By the elementary inequality, Cauchy–Schwarz inequality, the Assumptions1 and 2, we get(22)I1≤6ɛ2Γα2E∫0tt−sα−1fs,Xɛs,Xsɛ−fs,Yɛs,Ysɛds2+6ɛ2Γα2E∫0tt−sα−1fs,Yɛs,Ysɛ−f¯Yɛs,Ysɛds2≤6ɛ2tΓα2E∫0tt−s2α−2fs,Xɛs,Xsɛ−fs,Yɛs,Ysɛ2ds+6ɛ2tΓα2E∫0tt−s2α−2fs,Yɛs,Ysɛ−f¯Yɛs,Ysɛ2ds≤6ɛ2tsup0≤t≤TλtΓα2∫0tt−s2α−2EXɛr−Yɛr2+EXsɛ−Ysɛ2ds+6ɛ2t2Γα2sup0≤s≤tλ1tsup0≤s≤tEYɛs2+sup0≤s≤tEYsɛ2.
By the Itô isometry, the elementary inequality, Cauchy–Schwarz inequality, the Assumptions1 and 2, we get(23)I2≤3ɛΓα2E∫0tt−s2α−2gs,Xɛs,Xsɛ−g¯Yɛs,Ysɛℒ202ds≤6ɛΓα2E∫0tt−s2α−2gs,Xɛs,Xsɛ−gs,Yɛs,Ysɛℒ202ds+6ɛΓα2E∫0tt−s2α−2gs,Yɛs,Ysɛ−g¯Yɛs,Ysɛℒ202ds≤6ɛsup0≤t≤TλtΓα2∫0tt−s2α−2EXɛs−Yɛs2+EXsɛ−Ysɛ2ds+6ɛtΓα2sup0≤s≤tλ2ssup0≤s≤tEYɛs2+sup0≤s≤tEYsɛ2.
By Lemma1, elementary inequality, we have(24)I3≤3ɛ2HΓα2E∫0tt−sα−1σs,Xɛs,Xsɛ−σ¯Yɛs,YsɛdBsH2≤3CHɛ2Ht2H−1Γα2E∫0tt−s2α−2σs,Xɛs,Xsɛ−σ¯Yɛs,Ysɛℒ202ds≤6CHɛ2Ht2H−1Γα2E∫0tt−s2α−2σs,Xɛs,Xsɛ−σs,Yɛs,Ysɛℒ202ds+6CHɛ2Ht2H−1Γα2E∫0tt−s2α−2σs,Yɛs,Ysɛ−σ¯Yɛs,Ysɛℒ202ds≤6CHɛ2Ht2H−1sup0≤t≤TλtΓα2∫0tt−s2α−2EXɛs−Yɛs2+EXsɛ−Ysɛ2ds+6CHɛ2Ht2HΓα2sup0≤s≤tλ3ssup0≤s≤tEYɛs2+sup0≤s≤tEYsɛ2.
Submitting (22), (23), (24) to (21), we get(25)EXɛt−Yɛt2≤6ɛ2t+6ɛ+6CHɛ2Ht2H−1Γα2sup0≤t≤Tλt∫0tt−s2α−2EXɛs−Yɛs2+EXsɛ−Ysɛ2ds+6ɛ2t2+6ɛt+6CHɛ2Ht2HΓα2∑i=13sup0≤s≤tλissup0≤s≤tEYɛs2+sup0≤s≤tEYsɛ2.
Noting thatEXɛt−Yɛt2=0 when −τ≤t≤0, it reduces(26)EXɛt−Yɛt2≤12ɛ2t+12ɛ+12CHɛ2Ht2H−1Γα2sup0≤t≤Tλt∫0tt−s2α−2EXɛs−Yɛs2ds+6ɛ2t2+6ɛt+6CHɛ2Ht2HΓα2∑i=13sup0≤s≤tλissup0≤s≤tEYɛs2+sup0≤s≤tEYsɛ2.
By the Gronwall–Bellman inequality ([23]), we have(27)EXɛt−Yɛt2≤6ɛ2t2+6ɛt+6CHɛ2Ht2HΓα2∑i=13sup0≤s≤tλissup0≤s≤tEYɛs2+sup0≤s≤tEYsɛ2×∑k=0∞12ɛ2tα+1+12ɛtα+12CHɛ2Ht2H+α−1ksup0≤t≤TλtΓαkΓkα+1.
So,(28)sup−τ≤s≤tEXɛt−Yɛt2≤6ɛ2t2+6ɛt+6CHɛ2Ht2HΓα2∑i=13sup0≤s≤tλissup0≤s≤tEYɛs2+sup0≤s≤tEYsɛ2×∑k=0∞12ɛ2tα+1+12ɛtα+12CHɛ2Ht2H+α−1ksup0≤t≤TλtkΓαkΓkα+1.
So, we can selectβ∈0,1 and L>0, such that for every t∈0,Lɛ−β⊆0,T,(29)sup−τ≤t≤Lɛ−βEXɛt−Yɛt2≤Cɛ1−β,where(30)C=6L2ɛ1−β+6L+6L2HCHɛ2H−11−βΓα2×∑i=13sup0≤s≤Lɛ−βλissup0≤s≤Lɛ−βEYɛs2+sup0≤s≤Lɛ−βEYsɛ2×∑k=0∞12Lα+1ɛ2−αβ−β+12Lαɛ1−αβ+12L2H+α−1CHɛ2H−β2H+α−1ksup0≤t≤TλtkΓαkΓkα+1.
Therefore, for anyδ>0, there exists ε1∈0,ɛ0, such that for any ε∈0,ɛ1 and t∈0,Lɛ−β,(31)sup−τ≤t≤Lɛ−βEXɛt−Yɛt2≤δ.
The proof is completed.


## 4. Example

Let us consider the following FSDEs with delays:(32)Dtαxt=xt+xtt−12+1dWtdt+1dBtHdt,where t∈0,1,ft,xt,xt=xt+xtt−12,gt,xt,xt=1,σt,xt,xt=1,1/2<α<1.It is easy to verify thatf,g,σ satisfy the Assumptions 1 and 2. Define f¯x,y as follows:(33)∫01f¯x,y1−t2α−2dt=∫01ft,x,y1−t2α−2dt.We can derive thatf¯x,y=x1+2α−1/2α+1. Similarly, g¯x,y=1,σ¯x,y=1. The averaging form of (32) can be defined as(34)Dtαyt=yt1+2α−12α+1+1dWtdt+1dBtHdt.According to Theorem1, as ε goes to zero, the solutions xt and yt are equivalent in the sense of mean square. So, the results can be checked.



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