# Existence and Stability of Solutions of Fuzzy Fractional Stochastic Differential Equations with Fractional Brownian Motions

Authors: Elhoussain Arhrrabi; M’hamed Elomari; Said Melliani; Lalla Saadia Chadli
Journal: Advances in Fuzzy Systems
 (2021)

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## Abstract

The existence, uniqueness, and stability of solutions to fuzzy fractional stochastic differential equations (FFSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition are investigated. Finally, we investigate the exponential stability of solutions.


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## Body

## 1. Introduction

There appears to be confusion of various kinds in the modeling of several real world systems, such as trying to characterize a physical system and opinions on its parameters. To deal with this ambiguity, the fuzzy set theory will be used [1]. It is able to handle such linguistic statements mathematically using this theory, such as “large” and “less.” The capacity to investigate fuzzy differential equations (FDEs) in modeling numerous phenomena, including imprecision, is provided by a fuzzy set. In particular, the fuzzy stochastic differential equations (FSDEs), in instance, might be used to investigate a variety of economics and engineering problems that involve two types of uncertainty: randomness and fuzziness.The fuzzy Ito^ stochastic integral was powered in [2, 3]. In [4, 5], the fuzzy stochastic integral is driven by the Wiener process as a fuzzy adapted stochastic process. In [6], Fei et al. studied the existence and uniqueness of solutions to the (FSDEs) under non-Lipschitzian condition. In [7], Jafari et al. study FSDEs driven by fBm. Jialu Zhu et al., in [8], prove existence of solutions to SDEs with fBm. Ding and Nieto [9] investigated analytical solutions of multitime-scale FSDEs driven by fBm. Vas’kovskii et al. [10] prove that the pth moments, p≥1, of strong solutions of a mixed-type SDEs are driven by a standard Brownian motion and a fBm. Despite the fact that some research exists on the problem of the uniqueness and existence of solutions to SDEs and FSDEs which are disturbed by Brownian motions or semimartingales [4, 11–15], a kind of the FFSDEs driven by an fBm has not been investigated. Agarwal et al. [16, 17] considered the concept of solution for FDEs with uncertainty and some results on FFDEs and optimal control nonlocal evolution equations. Recently, Zhou et al., in [18–20], gave some important works on the stability analysis of such SFDEs. Our results are inspired by the one in [21] where the existence and uniqueness results for the FSDEs with local martingales under the Lipschitzian conditions are studied. The rest of this paper is given as follows. Section 2 summarizes the fundamental aspects. In Section 3, existence and uniqueness of solutions to the FFSDEs are proved. Moreover, the stability of solutions is studied in Section 4. Finally, in Section 5, a conclusion is given.


## 2. Preliminaries

This part introduces the notations, definitions, and background information that will be utilized throughout the article.LetKℝn be the family of nonempty convex and compact subsets of ℝn. In Kℝn, the distance dH is defined by(1)dHM,N=maxsupm∈Minfn∈Nm−n,supn∈Ninfm∈Mm−n,M,N∈Kℝn.We denote byℳΩ,A;Kℝn the family of A-measurable multifunction, taking value in Kℝn.Definition 1 (see [21, 22]).
A multifunctionG∈ℳΩ,A;Kℝn is called ℒp-integrably bounded if ∃h∈ℒpΩ,A,ℙ;ℝ+ such that G≤hℙ-a.e, where(2)B≔dHB,0^=supb∈Bb,forB∈Kℝn.
We denote by(3)ℒpΩ,A,ℙ;Kℝn≔G∈ℳΩ,A;Kℝn:G∈ℒpΩ,A,ℙ;ℝ+.
LetEn denote the set of the fuzzy x:ℝn⟶0,1 such that xα∈Kℝn, for every α∈0,1, where xα:=a∈ℝn:xa≥α, for α∈0,1, and x0:=cla∈ℝn:xa>0. Let the metric be d∞x,y≔supα∈0,1dHxα,yα, in En, a∈ℝ; we have d∞x+z,y+z=d∞x,y, d∞x+y,z+w≤d∞x,z+d∞y,w, and d∞ax,ay=ad∞x,y.Definition 2 (see [23]).
Letf:c,d⟶En; the fuzzy Riemann–Liouville integral of f is given by(4)Jc+αfu=1Γα∫cuu−vα−1fvdv.Definition 3 (see [23]).
LetDf∈Cc,d,En∩Lc,d,En. The fuzzy fractional Caputo differentiability of f is given by(5)DCc+αfu=Jc+1−αDfu=1Γ1−α∫cuu−v−αDfvdv.
Now, we define the Henry–Gronwall inequality [24], which can be used in the proof of our result.Lemma 1.
Letf, g:0,T⟶ℝ+ be continuous functions. If g is nondecreasing and there exists constants K≥0 and α>0 as(6)fu≤gu+K∫0uu−vα−1fvdv,u∈0,T,then(7)fu≤gu+∫0u∑m=1∞KΓαmΓmαu−vnα−1gvdv,u∈0,T.Ifgu=b is constant on 0,T, the previous inequality is transformed into(8)fu≤bEαKΓαuα,u∈0,T,where Eα is given by(9)Eαz=∑m=0∞zmΓmα+1.Remark 1 (see [24]).
For allu∈0,T, ∃NK∗>0 does not depend on b such that fu≤NK∗b.Definition 4 (see [21, 22]).
A functionf:Ω⟶En is said fuzzy random variable if fα is an A-measurable random variable ∀α∈0,1A fuzzy random variablef:Ω⟶En is said ℒp-integrably bounded, p≥1, if fα∈ℒpΩ,A,ℙ;Kℝn, ∀α∈0,1
LetℒpΩ,A,ℙ;En denote the set of all fuzzy random variables; they are ℒp-integrally bounded.
For the notion of an fBm, we referred to [25].
Let us define a sequence of partitions ofa,b by ψm,m∈ℕ such that ψm⟶0 as m⟶∞. If, in L2Ω,A,ℙ, ∑i=0m−1ϕtimBHti+1m−BHtim converge to the same limit for all this sequences ψm,m∈ℕ, then this limit is said a Stratonovich-type stochastic integral and noted by ∫abϕsdBHs. Let J:=0,T, where 0<T<∞.Definition 5 (see [21, 22]).
A functionf:J×Ω⟶En is called fuzzy stochastic process; if ∀t∈J, ft,.=ft:Ω⟶En is a fuzzy random variableA fuzzy stochastic processf is continuous; if f.,υ:J⟶En are continuous, and it is AtHt∈J-adapted if for every α∈0,1 and for all t∈J, ftα:Ω⟶Kℝn is AtH-measurableDefinition 6 (see [21, 22]).
The functionf is called measurable if fα:J×Ω⟶Kℝn is a ℬJ⊗A-measurable, for all α∈0,1The functionf:J×Ω⟶En is said to be nonanticipating if it is AtHt∈J-adapted and measurableRemark 2.
The processx is nonanticipating if and only if x is measurable with respect to N:=A∈ℬJ⊗A:Au∈AuH,u∈J, where, for u∈J, Au=υ:u,υ∈A.Definition 7 (see [21, 22]).
A fuzzy processf:J×Ω⟶En is said ℒp-integrally bounded if ∃h∈ℒpJ×Ω,N;ℝ/d∞fs,υ,0^≤hs,υ.
We denote byℒpJ×Ω,N;En the set of all ℒp-integrally bounded and nonanticipating fuzzy stochastic processes.Proposition 1 (see [4]).
Forf∈ℒpJ×Ω,N;En and p≥1, we have J×Ω∋t,υ⟶∫0tfs,υds∈ℒpJ×Ω,N;En and d∞-continuous.Proposition 2 (see [4]).
Forf,g∈ℒpJ×Ω,N;En and p≥1, we have(10)Esupa∈0,td∞p∫0afudu,∫0agudu≤tp−1∫0tEd∞pfu,gudu.Proposition 3 (see [26]).
Letψ:J⟶ℝn; then, for t∈J,(11)supa∈0,tE∫0aψsdBHs2≤ct,H∫0tψs2ds.
Let us define the embedding ofℝn to En as .:ℝn⟶En:(12)ra=1,ifa=r,0,ifa≠r.Proposition 4 (see [4]).
Assume that the functionψ:J⟶ℝn satisfies ∫0Tψv2dv<∞. Then,i The fuzzy stochastic Ito^ integral ∫0vψudBHu∈L2J×Ω,N;Enii For x∈L2J×Ω,N;En, we have, for u≤v∈J,(13)d∞∫0vxw1dw1+∫0vψw2dBHw2,∫0uxw1dw1+∫0uψw2dBHw2=d∞∫uvxw1dw1+∫uvψw2dBHw2,0^.


## 3. Main Result

Now, we investigate the FFSDEs driven by an fBm given by(14)DCαxs=Jℙ.1fs,xsds+gsdBHs,x0=ℙ.1x0,where(15)f:J×Ω×En⟶En,g:J⟶ℝn,x0:Ω⟶En,and BHss∈J is a fBm defined on Ω,A,AsHs∈J,ℙ with Hirst index H∈1/2,1.Definition 8.
A processx:J×Ω⟶En is said to be a solution to equation (14) if the following holds:ix∈L2J×Ω,N;En.iix is d∞-continuous.iii We have(16)xt=Jℙ.1x0+1Γα∫0tfs,xst−s1−αds+1Γα∫0tgst−s1−αdBHs.
We will assume that all through this paper,f:J×Ω×En⟶En is ℬdS⊗N|ℬd∞-measurable. Let the following assumptions be introduced.ℋ1 If x0 is A0-measurable, we have(17)Ed∞2x0,0^<∞.ℋ2 For fs,0^ and g, we have(18)maxd∞2fs,0^,0^,g≤c,for everys∈J.ℋ3 For all z,w∈En,(19)d∞2fs,z,fs,w≤cd∞2z,w,where c is equal to one in ℋ2.
Let us now introduce the main theorem in this part.Theorem 1.
Under assumptionsℋ1–ℋ3 and x0∈L2Ω,A0,ℙ;En, the equation (14) has a unique solution xt.Proof.
The method of successive approximations will be used to demonstrate the existence of a solution to (1). Therefore, define a sequence xn:J×Ω⟶En as follows:(20)x0t=ℙ.1x0,and for n=1,…,(21)xnt=Jℙ.1x0+1Γα∫0tfs,xn−1st−s1−αds+1Γα∫0tgst−s1−αdBHs.
It is clear thatxns are in L2J×Ω,N;En and d∞-continuous. Indeed, we have x0∈L2J×Ω,N;En and x0 is d∞-continuous.
Let us define forn∈ℕ and t∈JKn=sup0≤u≤tEd∞2xnu,xn−1u. Then, from Propositions 2 and 3 and ℋ1–ℋ3, we have(22)K1t=sup0≤u≤tEd∞21Γα∫0ufs,x0u−s1−αds+1Γα∫0ugsu−s1−αdBHs,0^≤2sup0≤u≤tEd∞21Γα∫0ufs,x0u−s1−αds,0^+1Γα∫0ugsu−s1−αdBHs2≤2sup0≤u≤t2Ed∞21Γα∫0ufs,x0u−s1−αds,1Γα∫0ufs,0^u−s1−αds+4sup0≤u≤tEd∞21Γα∫0ufs,0^u−s1−αds,0^+2Esupu∈0,tcT,HΓα∫0ugs2u−s1−αds≤4TΓα∫0tEd∞2fs,x0,fs,0^t−s1−αds+4TΓα∫0tEd∞2fs,0^,0^t−s1−αds+2cT,Ht−s1−α∫0tgs2t−s1−αds≤4TctαΓα+1d∞2x0,0^+4TtαcΓα+1+2c2cT,HtαΓα+1≔l1tαΓα+1,where l1=4cTd∞2x0,0^+4Tc+2c2cT,H. Moreover, similarly, we have(23)Kn+1t=sup0≤u≤tEd∞21Γα∫0ufs,xnsu−s1−αds+1Γα∫0ugsu−s1−αdBHs,1Γα∫0ufs,xn−1su−s1−αds+1Γα∫0ugsu−s1−αdBHs≤2sup0≤u≤tEd∞21Γα∫0ufs,xnsu−s1−αds,∫0ufs,xn−1su−s1−αds≤2tΓα∫0tt−sα−1Ed∞2fs,xns,fs,xn−1sds≤2tcΓα∫0tt−sα−1supu∈0,sEd∞2xns,xn−1sds≤2TcΓα∫0tt−sα−1Knsds.
Thus, we obtain(24)Knt≤l1l2l2tαnn!Γα+1,∀t∈J,n∈ℕ,where l2=2Tc.
Hence, from Chebyshev’s inequality and (24), we obtain(25)ℙsupu∈Jd∞2xnu,xn−1u>14n≤l1l24l2Tαnn!Γα+1,
Since the series∑n≥14l2Tαn/n! converges, according to Borel–Cantelli lemma, we obtain(26)ℙsupu∈Jd∞xnu,xn−1u>12n=0.
Thus, the sequencexn.,υ is uniformly convergent to x˜.,υ:J⟶ℝn for υ∈Ωc, where Ωc∈A and ℙΩc=1. Then,(27)limn⟶∞supt∈JEd∞2xnt,x˜t=0.
Let us definex:J×Ω⟶En as follows:(28)x.,υ=x˜.,υ,ifυ∈Ωc,freely chosen,ifυ∈υ∈ΩΩc.
We can observe that, for each0≤α≤1 and t∈J, we have(29)limn⟶∞dHxn,υα,xn−1,υα=0.
Then,xt,.α:Ω⟶Kℝn is At-measurable. Hence, x is nonanticipating. By (27), we have(30)limn⟶∞supt∈JEd∞2xnt,xt=0,which shows that ∃λ>0 independent of n∈ℕ such that(31)supt∈JEd∞2xnt,xt≤λ.
Sincexn∈L2J×Ω,N;En, we have xnt∈L2Ω,A,ℙ;En. In addition, we can prove that x∈L2J×Ω,N;En.
Indeed, for alln∈ℕ and t∈J, let us denote(32)ψnt=sup0≤u≤tEd∞2x0,0^.
Then, we obtain(33)ψnt≤3Ed∞2x0,0^+3sup0≤u≤tEd∞21Γα∫0uu−sα−1fs,xn−1sds,0^+3Esup0≤u≤t1Γα∫0uu−sα−1gsdBHs2.
By the triangle inequality,ℋ1–ℋ3, and Propositions 2 and 3, we have(34)ψnt≤3Ed∞2x0,0^+6tΓα∫0tt−sα−1Ed∞2fs,xn−1s,fs,0^+Ed∞2fs,0^,0^ds+3cT,HΓα∫0tt−sα−1gs2ds,≤3Ed∞2x0,0^+6ctΓα∫0tt−sα−1Ed∞2xn−1s,0^ds+6ctα+1Γα+1+3tαc2cT,HΓα+1.
We obtain(35)ψnt≤A1+A2∫0tt−sα−1ψn−1sds,where A1=3Ed∞2x0,0^+6ctα+1/Γα+1+3tαc2cT,H/Γα+1 and A2=6ct/Γα.
According to Lemma1 and Remark 1, there exist a constant MA2>0 independent of A1 such that(36)ψnt≤MA2A1.
Due toℋ1, (31), and (36), we obtain(37)sup0≤s≤tEd∞2xs,0^≤2sup0≤s≤tEd∞2xs,xns+2sup0≤s≤tEd∞2xns,0^,≤2λ+2MA2A1<∞,which implies(38)∫0TEd∞2xs,0^ds≤Tsupt∈JEd∞2xt,0^<∞.
Thus, we getx∈L2J×Ω,N;En.
On the contrary, we have(39)supt∈JEd∞2xt,x0+1Γα∫0tfs,xst−s1−αds+1Γα∫0tgst−s1−αdBHs=0.
Indeed, we observe(40)supt∈JEd∞2xt,x0+1Γα∫0tfs,xst−s1−αds+1Γα∫0tgst−s1−αdBHs≤3supt∈JEd∞2xt,xnt+supt∈JEd∞2xnt,x0+1Γα∫0tfs,xn−1st−s1−αds+1Γα∫0tgst−s1−αdBHs+supt∈JEd∞2x0+1Γα∫0tfs,xn−1st−s1−αds+1Γα∫0tgst−s1−αdBHs,x0+1Γα∫0tfs,xst−s1−αds+1Γα∫0tgst−s1−αdBHs≔I1+I2+I3,where limn⟶∞I1=0 and I2=0. For I3, by using Propositions 2 and 3, ℋ3, and (30), we have(41)limn⟶∞I3≤limn⟶∞Tα+1cΓα+1supt∈JEd∞2xu,xn−1udu=0.
Hence, we get (39), which implies (16) holds. Hence, from definition (8), xt is a solution to equation (14).
For the uniqueness of a solutionx, suppose that x,z:J×Ω⟶En are solutions to equation (14). We denote by Kt≔supv∈JEd∞2xv,zv. So, for each t∈J, we obtain(42)Kt≤tcΓα∫0tEd∞2xs,zst−s1−αds≤TcΓα∫0tKst−s1−αds.
Thus, by Lemma1, we have, for t∈J, Kt≡0, which implies(43)supt∈Jd∞xt,zt=ℙ.10.


## 4. Stability Result

In this part, we examine the stability of the solution with respect to initial values by using Henry–Gronwall inequality. Indeed, letx and z denote the solutions of the following FFSDEs:(44)CDαxs=Jℙ.1fs,xsds+gsdBHs,x0=ℙ.1x0,(45)CDαzs=Jℙ.1fs,xsds+gsdBHs,z0=ℙ.1x0,respectively.Proposition 5.
Suppose thatx0,z0∈L2Ω,A0,ℙ;En and f:J×Ω×En⟶Eng:J⟶ℝn satisfy ℋ1–ℋ3. Then,(46)sup0≤u≤tEd∞2xu,zu≤λ0Mλ1,where λ0=2Ed∞2x0,z0 and λ1=2Tc/Γα. Especially, xt=Jℙ.1zt if x0=ℙ.1z0.Proof.
Suppose thatx,z:J×Ω⟶En are solutions to equations (44) and (45), respectively. So, let Kt≔Esup0≤u≤td∞2xu,zu. Due to Propositions 2 and 3 and ℋ3, we obtain(47)Kt≤2Ed∞2x0,z0+2Γαsupu∈0,tEd∞2∫0ufs,xst−s1−αds,∫0ufs,zst−s1−αds≤2Ed∞2x0,z0+2TcΓα∫0tEd∞2xs,zst−s1−αds≤2Ed∞2x0,z0+2TcΓα∫0tsupEu∈0,sd∞2xu,zut−s1−αdu=2Ed∞2x0,z0+2TcΓα∫0tKst−s1−αds≔λ0+λ1∫0tKst−s1−αds.
Then, according to Lemma1 and Remark 1, there exist a constant Mλ1>0 independent of λ0 such that(48)Kt≤λ0Mλ1,∀t∈J.
Then,λ0=0 if x0=ℙ.1z0. Therefore, we know that xt=Jℙ.1zt.
Finally, we examine the exponential stability of solutions to the FFSDEs which disturbed an fBm with respect tof and g. Thus, let x and xn denote solutions to the following FFSDEs:(49)CDαxs=Jℙ.1fs,xsds+gsdBHs,x0=ℙ.1x0,(50)CDαxns=Jℙ.1fns,xnsds+gnsdBHs,xn0=ℙ.1x0,respectively.Proposition 6.
Suppose thatx0∈L2Ω,A0,ℙ;En and f,fn:J×Ω×En⟶Eng,gn:J⟶ℝnn∈ℕ fulfill ℋ1–ℋ3. Furthermore, assume that(51)limn⟶∞1Γα∫0tt−sα−1Ed∞2t,x,fnt,xt,xdt=0,(52)limn⟶∞1Γα∫0tt−sα−1gns−gs2ds=0.
Then, we have(53)limn⟶∞Esupt∈Jd∞2xt,xnt=0,where x,xn:J×Ω⟶En are solutions of equations (49) and (50), respectively.Proof.
According to Theorem1, the solutions x and xn are unique and exist. From Propositions 3 and 4, we deduce that, for every t∈J,(54)sup0≤u≤tEd∞2xu,xnu≤2sup0≤u≤tEd∞21Γα∫0ufs,xsu−s1−αds,1Γα∫0ufns,xnsu−s1−αds+2sup0≤u≤tEd∞21Γα∫0ugsu−s1−αdBHs,1Γα∫0ugnsu−s1−αdBHs≤q4sup0≤u≤tEd∞21Γα∫0ufns,xsu−s1−αds,1Γα∫0ufns,xnsu−s1−αds+4sup0≤u≤tEd∞21Γα∫0ufs,xsu−s1−αds,1Γα∫0ufns,xsu−s1−αds+2cT,HΓα∫0tgs−gns2t−s1−αds≤4ctΓα∫0tEd∞2xs,xnst−s1−αds+4tΓα∫0tt−sα−1Ed∞2fs,xs,fns,xsds+2cT,HΓα∫0tgs−gnss2t−s1−αds≤β1n+β2∫0tsup0≤u≤sEd∞2xu,xnst−s1−αds,where(55)β1n≔4TΓα∫0tEd∞2fs,xs,fns,xst−s1−αds+2cT,HΓα∫0tgs−gns2t−s1−αds,β2=4cT/Γα. From Lemma 2 and Remark 1, ∃Mβ2>0 is independent of β1n such that(56)supu∈0,tEd∞2xu,xnu≤β1nMβ2.
Hence, from (51) and (52), we get limn⟶∞β1n=0.


## 5. Conclusions

In this study, we have proved the existence and uniqueness of solutions to FFSDEs under the Lipschitzian coefficient. On the contrary, the stability of solutions to the FFSDEs is analyzed.



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