# An Averaging Principle for Mckean–Vlasov-Type Caputo Fractional Stochastic Differential Equations

Authors: Weifeng Wang; Lei Yan; Junhao Hu; Zhongkai Guo
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we want to establish an averaging principle for Mckean–Vlasov-type Caputo fractional stochastic differential equations with Brownian motion. Compared with the classic averaging condition for stochastic differential equation, we propose a new averaging condition and obtain the averaging convergence results for Mckean–Vlasov-type Caputo fractional stochastic differential equations.


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## Body

## 1. Introduction

For complex systems, we usually want to locate an effective simplified model to approximate the original complex system or extract the main dynamical behavior of the original system. Based on these ideas, a lot of effective methods have been generated in dynamical systems, such as invariant manifolds, averaging principle, and homogenization principle. These effective methods have now been extended to deal with stochastic systems, such as stochastic invariant manifolds see [1, 2] and stochastic averaging principle, see [3–9].Currently, the problem of averaging for stochastic differential equations have received a lot of attention and various types of stochastic differential equations have been studied, see [4, 6, 7, 10–12]. However, there are no relevant results of averaging principle for distribution dependent-type stochastic differential equations which we will consider in this paper.On the contrary, the problem of averaging for stochastic fractional order differential equations have received a lot of attention in recent years, and some results [13] have been obtained under averaging condition consistence with the classic case (see [4, 5, 14]). Noting that the fractional order derivative is a nonlocal operator, therefore, the fractional order differential equation is more effective for describing certain phenomena in the real world (see [15–17]). Current research studies on stochastic fractional order differential equations mainly focused on the existence and uniqueness of the solutions, with fewer results from the dynamical system perspective.Based on the above discussion, we shall study the averaging principle for the following Mckean–Vlasov-type Caputo fractional stochastic differential equations:(1)DtαXt=ft,Xt,μtdt+gt,Xt,μtdBt,t≥0,μt=probability distribution ofXt,X0=x0∈L2Ω,H,where α∈1/2,1 and Bt is a scalar Brownian motion. Nonlinear terms f and g are H-valued functions defined on R+×H×Mγ2H, and Mγ2H denotes a proper subset of probability measure on H. If the terms f and g do not depend on the probability distribution μt of the process X at time t, such equations have been studied by [13] and other authors. If α=1, the equation becomes a classical Mckean–Vlasov-type stochastic differential equations which have been considered by many authors with different approaches (see [18–20]). In this paper, we just focused on α∈1/2,1, and more details can be seen in Section 2.The paper is structured as follows. We introduce some notation and assumptions in Section2. The existence and unique solution for distribution dependent fractional stochastic differential equations will be discussed in Section 3. An averaging principle for the above equation is established in Section 4.


## 2. Preliminaries

First, we introduce some notation. LetCH be the space of continuous functions on H. Let ℬH be the Borel σ-algebra of subsets of H. MH is the space of probability measures on ℬH and carries the usual topology of weak convergence. μ,ϕ denotes ∫Hϕxμdx. Let γx=1+x, ∀x∈H, and then, define the Banach space(2)CρH=ϕ∈CH:ϕCρ≡supx∈Hϕxγ2x+supx≠yϕx−ϕyx−y<∞.For anyp≥1, let MγpsH denote the Banach space of signed measures m on H, and μγp≡∫Hγpmdx<∞. m=m++m− and m=m+−m− are the Jordan decomposition of m. Mγ2H=MγpsH∩MH is the set of probability measures on ℬH, and there exists second moments. Define the following metric:(3)ρμ,ν=supϕ,μ−ν:ϕρ=supx∈Hϕxγ2x+supx≠yϕx−ϕyx−y≤1.Then,Mγ2H,ρ is a complete metric space. Let C0,T,Mγ2H,ρ be the complete metric space of continuous functions from 0,T to Mγ2H,ρ with the following metric:(4)DTμ,ν=supt∈0,Tρμt,νt,forν,μ∈C0,T,Mγ2H,ρ.More details can be seen in [18].In order to obtain the existence and uniqueness of the solution of (1), we introduce the following conditions.(i)
H1 (Lipschitz condition): for all x,y∈H and t∈0,T, μ∈C0,T,Mγ2H,ρ, and there exists a bounded function k1t>0, such that(5)ft,x,μ−ft,y,ν2+gt,x,μ−gt,y,ν2≤k1tx−y2+ρ2μ,ν.(ii)
H2 (growth condition): for all x,t∈H×0,T, there exists a bounded function k2t>0 such that(6)ft,x,μ2+gt,x,μ2≤k2t1+x2.In this paper, we assume there existence of a constantk such that maxk1t,k2t≤k.First, we give an important lemma, which is a type of promotion form of Gronwall’s inequality with singular kernels.Lemma 1 (see [21, 22]).
Supposeb≥0, β>0, and at is a nonnegative function locally integrable on 0≤t<T (some T≤+∞), and suppose ut is nonnegative and locally integrable on 0≤t<T with(7)ut≤at+b∫0tt−sβ−1usds,on this interval. Then,(8)ut≤at+∫0t∑n=1∞bΓβnΓnβt−snβ−1asds,0≤t<T.


## 3. Existence and Uniqueness

Consider the integral form of equation (1):(9)Xt=x0+1Γα∫0tt−sα−1fs,Xs,μtds+1Γα∫0tt−sα−1gt,Xs,μtdBt.Under the assumptions ofH1 and H2, we will prove the existence and uniqueness of solution for the above equation.Definition 1.
Anℱt-adapted stochastic process Xt with law LXt=μt is called a solution of (1) if Xt is continuous, and for ∀t∈0,T with X0=x0,(10)Xt=x0+1Γα∫0tt−sα−1fs,Xs,μtds+1Γα∫0tt−sα−1gt,Xs,μtdBt,ℙ−a.s.Theorem 1.
Assume that H1 and H2 hold; then, for∀x0∈L2Ω,H, equation (10) has a unique solution Xt∈C0,T;L2Ω,H with the associate probability distribution μt=LXt,t∈0,T belonging to C0,T,Mγ2X,ρ, such that(11)sup0≤t≤TEXt2<∞.We will proof the theorem by several steps.(i)
First, we prove thatXt∈L∞0,T,L2Ω;H for ∀μ∈C0,T,Mγ2H,ρ. Using the following inequality,(12)a+b+c2≤3a2+b2+c2.We see that(13)EXt2≤3Ex02+3E1Γα∫0tt−sα−1fs,Xs,μsds2+3E1Γα∫0tt−sα−1gt,Xs,μsdBt2≔3I1+3I2+3I3.ForI2, applying Cauchy–Schwarz’s inequality and H2, it follows(14)I2≤TkΓα2∫0tt−s2α−21+EXs2+μtγ2ds≤TkΓα2t2α−12α−11+sup0≤t≤Tμtγ2+∫0tt−s2α−2EXs2ds≤kT2αΓα22α−11+sup0≤t≤Tμtγ2+TkΓα2∫0tt−s2α−2EXs2ds.ForI3, by It o^’s isometry formula and H2, we have(15)I3≤kΓα2∫0tt−s2α−21+EXs2+μtγ2ds≤kT2α−1Γα22α−11+sup0≤t≤Tμtγ2+kΓα2∫0tt−s2α−2EXs2ds.Combining the above estimate results, we finally obtain(16)EXt2≤r1+r2∫0tt−s2α−1−1EXs2ds,where(17)r1≔3Ex02+3kT2α−1T+1Γα22α−11+sup0≤t≤Tμtγ2,r2≔3kT+1Γα2.With the help of Lemma1, it follows(18)EXt2≤r11+∫0t∑n=1∞r2Γ2α−1nΓ2α−1nt−s2α−1n−1ds≤r11+∑n=1∞r2Γ2α−1T2α−1nΓ2α−1n+1=r11+E2α−1,1r2Γ2α−1T2α−1<∞,for∀t∈0,T, and E2α−1,1⋅ is a two-parameter function of the Mittag-Leffler type [21].Then,(19)sup0≤t≤TEXt2<∞,andXt∈L∞0,T,L2Ω;H.(ii)
Now, we show thatXt∈C0,T,L2Ω;H for ∀μ∈C0,T,Mγ2H,ρ:(20)EXt−Xt02≤2E1Γα2∫0tt−sα−1fs,Xs,μsds−∫0t0t0−sα−1fs,Xs,μsds2+2E1Γα2∫0tt−sα−1fs,Xs,μsdBs−∫0t0t0−sα−1fs,Xs,μsdBs2≕2J1+J2.ForJ1, we have(21)J1≤2E1Γα2∫t0tt−sα−1fs,Xs,μsds2+2E1Γα2∫0t0t−sα−1−t0−sα−1fs,Xs,μsds2≕2J11+2J12.By the Cauchy–Schwartz inequality,(22)J11≤1Γα2k∫t0tt−s2α−2ds∫t0tEfs,Xs,μs2ds≤kΓα22α−1t−t02α−1∫t0t1+EXs2+μs2ds≤CkΓα22α−1t−t02α.ForJ12, we have(23)J12=E1Γα2∫0t0t−sα−1−t0−sα−1fs,Xs,μsds2≤kΓα2∫0t0t−sα−1−t0−sα−12ds∫0t01+EXs2+μs2ds≤CTkΓα2∫0t0t0−s2α−1−t−s2α−1ds≤CTkΓα2t−t02α−12α−1+t02α−12α−1−t2α−12α−1≤CTkΓα2t−t02α−12α−1.ForJ2, using the It o^ isometry formula, in the similar way as J1, we can prove that(24)J2≤CkΓα2t−t02α−12α−1.Results ofJ1 and J2 combined together show that(25)EXt−Xt02≤Ck1+TΓα2t−t02α−12α−1+CkΓα22α−1t−t02α,which impliedXt∈C0,T,L2Ω;H for each fixed μ∈C0,T,Mγ2,ρ.(iii)
By virtue of the fixed point theorem for contraction mappings, we can show that, for each fixedμ∈C0,T,Mγ2,ρ, equation (10) has a unique solution in C0,T,L2Ω;H. Similar arguments are also discussed in [18]. Now, we define an operator Φμ⋅ on C0,T,L2Ω;H:(26)ΦμXt=x0+1Γα∫0tt−sα−1fs,Xs,μsds+1Γα∫0tt−sα−1gt,Xs,μsdBt.It is easy to verify thatΦμ is from C0,T,L2Ω;H into itself.ForXt,Yt∈C0,T,L2Ω;H with x0=y0, let ⋅ζ denote the norm of C0,T,L2Ω;H and β=2α−1>0; then, we obtain(27)ΦμXt−ΦμYtς2≤21Γα∫0tt−sα−1fs,Xs,μs−fs,Ys,μsdsς2+21Γα∫0tt−sα−1gt,Xs,μs−gt,Ys,μsdBtς2.Using the Cauchy–Schwartz inequality and Ito^ isometry formula, it is readily seen that(28)ΦμXt−ΦμYtς2≤2Tk+2kΓα2∫0tt−sβ−1Xs−Ysς2ds.Our goal is now to prove the following inequality:(29)ΦμnXt−ΦμnXtς2≤1β2Tk+2kΓα2nΓβnΓnβtnβXt−Ytς2.The proof is based on mathematical induction overn. For n=1,(30)ΦμXt−ΦμYtς2≤2Tk+2kΓα2Xt−Ytς2tββ,which is fulfilled.For the induction step fromn = l to n = l + 1, we assume that, for n = l, equation (29) is satisfied; then,(31)Φμl+1Xt−Φμl+1Ytς2≤2Tk+2kΓα2∫0tt−sβ−1ΦμlXs−ΦμlYsς2ds≤2Tk+2kΓα2∫0tt−sβ−11β2Tk+2kΓα2lΓβlΓkβslβXs−Ysς2ds≤2Tk+2kΓα2l+11βΓβkΓlβXt−Ytς2∫0tt−sβ−1slβds.Thus, we only need to discuss the following integral:(32)∫0tt−sβ−1slβds.Lets=tz; then,(33)∫0tt−sβ−1slβds=∫011−zβ−1tβ−1tlβzlβtdz=tl+1β∫011−zβ−1zlβdz=tl+1βBlβ+1,β=tl+1βΓβΓlβ+1Γl+1β+1,whereB⋅,⋅ is the Beta function. Substitute the above equality into (31), and we derive that(34)Φμl+1Xt−Φμl+1Ytς2≤2Tk+2kΓ2αl+11βΓβlΓlβXt−Ytς2tl+1βΓβΓlβ+1Γl+1β+1=2Tk+2kΓ2αl+11βΓβl+1Γlβ+1Γl+1β+1Γlβtl+1βXt−Ytς2=2Tk+2kΓ2αl+11βΓβl+1lβΓlβl+1βΓl+1βΓlβtl+1βXt−Ytς2≤2Tk+2kΓ2αl+11βΓβl+1tl+1βΓl+1βXt−Ytς2.By the above discussion, we finally obtain(35)ΦμnXt−ΦμnYtς2≤2Tk+2kΓ2αn1βΓβnTnβΓnβXt−Ytς2.Note thatTk+k/Γαn1/βΓβnTnβ/Γnβ⟶0 as n⟶+∞. Then, we have Tk+k/Γαn1/βΓβnTnβ/Γnβ<1 for any sufficiently large n. And, this shows that Φμ⋅ is a contraction map on C0,T,L2Ω;H. So, it has a unique fixed point for μ∈C0,T,Mγ2H,ρ.(iv)
LXμ=LXμt:t∈0,T is the probability law of Xμ. Now, we prove that LXμ∈C0,T,Mγ2,ρ. Notice that Xμ∈C0,T,L2Ω;H,LXμ∈Mγ2H for ∀t∈0,T. So, we only need to prove t⟶LXμt is continuous.In step (ii), we have(36)EXμt−Xμs2⟶0,ast⟶s.By the definition ofρ, we have(37)ϕ,LXμt−LXμs=EϕXμt−ϕXμs≤ϕρEXμt−Xμs≤ϕρEXμt−Xμs21/2,which implies(38)limt⟶sρLXμt,LXμs=0.Hence, we verify thatLXμ∈C0,T,Mγ2,ρ.(v)
DefineΨ on C0,T,Mγ2,ρ as follows:(39)Ψ:μ⟶LXμ.In the following, we will show that the operatorΨ has a unique fixed point in C0,T,Mγ2,ρ. Take μ,ν∈C0,T,Mγ2,ρ, and let Xμt and Xνt be the corresponding solutions of the following equations:(40)Xμt=x0+1Γα∫0tt−sα−1fs,Xs,μsds+1Γα∫0tt−sα−1gt,Xs,μsdBt,Xνt=x0+1Γα∫0tt−sα−1fs,Xs,νsds+1Γα∫0tt−sα−1gt,Xs,νsdBt.Thus,(41)EXμt−Xνt2≤2TΓα2E∫0tt−s2α−2fs,Xμs,μs−fs,Xνs,νs2ds+2Γα2E∫0tt−s2α−2gs,Xμs,μs−gs,Xνs,νs2ds≤2kTΓα2E∫0tt−s2α−2Xμs−Xνs2+ρ2μs,νsds+2kΓα2E∫0tt−s2α−2Xμs−Xνs2+ρ2μs,νsds.After simple calculation, we have that(42)sup0≤t≤TEXμt−Xνt2≤2kT+1Γα2T2α−12α−1sup0≤t≤TEXμt−Xνt2+DT2μ,ν.Select the appropriateT=T0>0, such that(43)2kT0+1Γα2T02α−12α−1<13.Then, it follows(44)sup0≤t≤T0EXμt−Xνt2<12DT02μ,ν.By the definition ofρμ,ν and DT2μ,ν, we can obtain(45)ρ2μ,ν≤EXμt−Xνt2.Taking sup-norm on both sides, we obtain(46)DT2Ψμ,Ψν≤sup0≤t≤T0EXμt−Xνt2.Combine this result with equation (44), and we finally derive(47)DT02Ψμ,Ψν<12DT02μ,ν.SinceΨ is a contraction in C0,T0,Mγ2H,ρ, it has a unique fixed point. Thus, equation (10) has a unique solution Xt with μ=LXt on 0,T0. Because Xt belongs to C0,T,L2Ω;H, we can extend the solution to 0,T by considering 0,T0,T0,2T0, and so on. This completes the proof.


## 4. An Averaging Principle

In this section, we study an averaging principle for the following distribution dependent fractional stochastic differential equations inH:(48)Xϵt=x0+ϵΓα∫0tt−sα−1fs,Xϵs,μϵsds+ϵΓα∫0tt−sα−1gs,Xϵs,μϵsdBt,where x0∈L2Ω;H. We will show that the solution of (48) will be approximated by the following simpler or averaged process under certain conditions:(49)Zϵt=x0+ϵΓα∫0tt−sα−1f¯Zϵs,νϵsds+ϵΓα∫0tt−sα−1g¯Zϵs,νϵsdBt.Equation (49) is called the averaged equation for (48). Now, we prove that the solution of (49) converges to the solution of the original equation (48) under the following additional conditions.H3:(50)1t2α−1∫0tt−s2α−1fs,x,μ−f¯x,μ2ds≤φ1t1+x2+μ2.H4:(51)1t2α−1∫0tt−s2α−1gs,x,μ−g¯x,μ2ds≤φ2t1+x2+μ2,where φit are positive and bounded with limt⟶+∞φit=0 for i=1,2.Remark 1.
Note that when we takeα=1, then this condition is consistence with the classic case, see [4].Theorem 2.
LetH1−H4 hold. Then, for ∀δ1>0, there exist constants L>0, ϵ1∈0,ϵ0 and β∈0,1 such that, for any ϵ∈0,ϵ1, 1/2<α<1, we have(52)sup0≤t≤Lε−β/2α−1EXεt−Zεt2≤δ1.Let us consider(53)Xϵt−Zϵt=ϵΓα∫0tt−sα−1fs,Xϵs,μϵs−f¯Zϵs,νϵsds+ϵΓα∫0tt−sα−1gs,Xϵs,μϵs−g¯Zϵs,νϵsdBt.By the arithmetic inequality, it follows that(54)EXϵt−Zϵt2≤2EϵΓα∫0tt−sα−1fs,Xϵs,μϵs−f¯Zϵs,νϵsds2+EϵΓα∫0tt−sα−1gs,Xϵs,μϵs−g¯Zϵs,νϵsdBt2≕2Q1+Q2.ForQ1, we have(55)Q1=EϵΓα∫0tt−sα−1fs,Xϵs,μϵs−f¯Zϵs,νϵsds2≤2ϵ2Γα2E∫0tt−sα−1fs,Xϵs,μϵs−fs,Zϵs,νϵsds2+E∫0tt−sα−1fs,Zϵs,νϵs−f¯Zϵs,νϵsds2.Applying the Cauchy–Schwarz inequality,H1, H3, and the definition of the metric ρ, we obtain(56)Q1≤2tkϵ2Γα2∫0tt−s2α−2EXϵs−Zϵs2+ρ2μϵs,νϵsds+2ϵ2Γα2t2α1t2α−1E∫0tt−s2α−2fs,Zϵs,νϵs−f¯Zϵs,νϵs2ds≤4tkϵ2Γα2∫0tt−s2α−2EXϵs−Zϵs2ds+2ϵ2Γα2t2αφ1t1+EZϵs2+νϵs2.ForQ2, using the Ito^ isometry formula, we obtain(57)Q2=ϵΓα2∫0tt−s2α−2Egs,Xϵs,μϵs−g¯Zϵs,νϵs2ds≤2ϵΓα2∫0tt−s2α−2Egs,Xϵs,μϵs−gZϵs,νϵs2ds+∫0tt−s2α−2EgZϵs,νϵs−g¯Zϵs,νϵs2ds.Applying conditionsH1 and H4, we derive(58)Q2≤2kϵΓα2∫0tt−s2α−1EXϵs−Zϵs2+ρ2μϵs,νϵsds+∫0tt−s2α−1EgZϵs,νϵs−g¯Zϵs,νϵs2ds≤4kϵΓα2∫0tt−s2α−1EXϵs−Zϵs2ds+2ϵΓα2t2α−1φ2t1+EZϵs2+νϵs2.Therefore, from the above discussion, (56)–(58), and Theorem 1, we have(59)EXϵt−Zϵt2≤4tkϵ2Γα2∫0tt−s2α−1EXϵs−Zϵs2ds+2ϵ2Γα2t2αφ1t1+EZϵs2+νϵs2+4kϵΓα2∫0tt−s2α−1EXϵs−Zϵs2ds+2ϵΓα2t2α−1φ2Γα21+EZϵs2+νϵs2=2ϵT2α−1c1ϵΓα2T+c2Γα2+4ϵkϵΓα2T+kΓα2∫0tt−s2α−1−1EXϵs−Zϵs2ds.Denoter1=2c1ϵ/Γα2T+c2/Γα2 and r2=4kϵ/Γα2T+k/Γα2; using Lemma 1, we have(60)EXϵt−Zϵt2≤ϵT2α−1r11+∫0t∑n=1∞r2ϵΓ2α−1nΓ2α−1nt−s2α−1n−1ds≤ϵT2α−1r11+∑n=1∞r2ϵΓ2α−1T2α−1nΓ2α−1n+1≤ϵT2α−1r11+E2α−1,1r2ϵΓ2α−1T2α−1.Select someβ∈0,1, L>0, such that, for ∀t∈0,Lϵ−β/2α−1, we obtain(61)sup0≤t≤Lϵ−β/2α−1EXϵt−Zϵt2≤Cϵ1−β,where C=r11+E2α−1,1r2Lϵ−β/2α−1Γ2α−1.Consequently, for∀δ1>0, one can select some ϵ1∈0,ϵ0 such that, for each ϵ∈0,ϵ1, ∀t∈0,Lϵ−β/2α−1, we have(62)sup0≤t≤Lϵ−β/2α−1EXϵt−Zϵt2≤δ1.This completes the proof.Remark 2.
Using the definition ofρ, we obtain(63)ρμϵt,νϵt≤EXϵt−Zϵt≤EXϵt−Zϵt2.
From the above estimate, we actually obtain(64)DTμϵ,νϵ=sup0≤t≤Tρμϵt,νϵt≤ssup0≤t≤TEXϵt−Zϵt2⟶0,ϵ⟶0,which means that, as ϵ⟶0, μϵt corresponding to Xϵt converges to νϵt of Zϵt in C0,T;Mγ2,ρ.



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