# Fractional Brownian Motion for a System of Fuzzy Fractional Stochastic Differential Equation

Authors: Kinda Abuasbeh; Ramsha Shafqat
Journal: Journal of Mathematics
 (2022)

---

## Abstract

We study fractional Brownian motion– (FBM–) driven fuzzy stochastic fractional evolution equations. These equations can be used to model fuzziness, long-range dependence, and unpredictability in hybrid real-world systems. Under various assumptions regarding the coefficients, we investigate the existence-uniqueness of the solution using an approximation method to the fractional stochastic integral. We can solve an equation with linear coefficients, for example, in financial models Application to a model of population dynamics is also illustrated. An example is propounded to show the applicability of our results.


---

## Body

## 1. Introduction

Fractional stochastic differential equations (FSDEs) play an important role in the modeling of numerous complicated processes in several sectors of science and engineering. FSDE theory and applications were examined. Furthermore, numerous academics have produced interest in systems with memory or aftereffects.There appears to be some difficulty in modeling a variety of modern-world systems, such as trying to characterize the physical system and differing viewpoints on its properties. The fuzzy set theory will be utilized to resolve this issue [1]. It can handle linguistic claims like “big” and “less” mathematically using this approach. A fuzzy set provides the ability to examine fuzzy differential equations (FDEs) in representing a variety of phenomena, including imprecision. For example, fuzzy stochastic differential equations (FSDEs) could be used to explore a wide range of economic and technical problems involving two types of uncertainty: randomness and fuzziness.Deterministic fuzzy differential equations were developed as a result of research into dynamic systems with inadequate or ambiguous parameter information. They are increasingly used in system models in biology, engineering, civil engineering, bioinformatics, and computational biology, quantum optics and gravity, hydraulic and mechanical system modeling. Many studies in this field have been conducted utilizing various ways of expressing differential problems in a fuzzy framework. The Hukuhara derivative of a set-valued function was used as the foundation for the first approach to deterministic FDEs.There are several articles on FSDEs, each of which takes a different approach. The fuzzy stochastic ItÃ´ integral was defined by the author in [2]. The fuzzy Ito^ stochastic integral was driven by fuzzy non-anticipating stochastic processes and the Wiener process. Malinowski [3–5] worked on the application, properties, Ito type strong solutions, and with delay to stochastic fuzzy differential equations. To construct a fuzzy random variable, the method involves embedding crisp Ito^ stochastic integral into fuzzy space. Guo et al. [6], Li et al. [7], Deng et al. [8], and Ahmad [9] worked on the stability of fractional stochastic differential equation. Hussain et al. [10] studied stochastic modeling of COVID-19. Abbas et al. [11, 12] solve ordinary differential equation. In the study of Niazi et al. [13], Iqbal et al. [14], Shafqat et al. [15], Alnahdi [16], Khan [17], and Abuasbeh et al. [18–21], existence and uniqueness of the fuzzy fractional evolution equations were investigated.FBM has been used to describe the behavior of asset prices and stock market volatility. This process is a good fit for describing these values because of its long-range dependence on self-similarity qualities. For a general discussion of the applications of FBM to model financial quantities, see Shiryaev [22]. Several writers have proposed a fractional Black and Scholes model to replace the traditional Black and Scholes model, which is memoryless and depends on the so-called fractional Black and Scholes model of geometric Brownian motion. The risky asset’s market stock price is given by this model:
(1)Sν=S0expμν+σBνH−σ22ν2H,where BH is an FBM with the Hurst parameter H,μ is the mean rate of return, and σ>0 is the volatility, and at time ν, the price of non-risky assets is erν, where r is the interest rate.In modeling many stochastic systems, on the other hand, the FBM, which exhibits long-range dependency, is proposed to replace Brownian motion as a driving mechanism. The FBM is a Gaussian process with favorable qualities such as long-range dependency, self-similarity, and increment stationarity whenH∈1,2 is used as the Hurst parameter. This method is well suited to the study of phenomena with long-range and scale-invariant correlations. When H≠3/2, however, FBM is not semimartingale.Jafari et al. [23] worked on FSDEs driven by FBM. Inspired by the above paper, we introduce fuzzy fractional stochastic differential equation (FFSDEs) in relation to FBM in this study for order (1,2). These equations can be used to simulate unpredictability, fuzziness, and long-range dependence in hybrid dynamic systems. To determine the explicit answers, we use an approximation approach to fractional stochastic integrals. To investigate existence-uniqueness of strong solutions, we use Liouville form of FBM with parameter H∈3/2,2. Furthermore, we discuss using the equations in financial models:
(2)Dc0ναXν=fν,Xν,DβXνds+gν,XνDβXνdBHν,ν∈0,I,X0+mX=X0,X′0=X1,where α∈1,2, β∈0,1, and BH are FBM and f,g:0,I⟶Rm is the continuous function.There has been a recent interest in input noises lacking independent increments and exhibiting long-range dependence and self-similarity qualities, which has been motivated by some applications in hydrology, telecommunications, queueing theory, and mathematical finance. When the covariances of a stationary time series converge to zero like a power function and diverge so slowly that their sums diverge, this is known as long-range dependence. The self-similarity property denotes distribution invariance when the scale is changed appropriately. FBM, a generalization of classical Brownian motion, is one of the simplest stochastic processes that are Gaussian, self-similar, and exhibit stationary increments. When the Hurst parameter is more than 1/2, the FBM exhibits long-range dependency, as we will see later. In this note, we look at some of the features of FBM and discuss various strategies for constructing a stochastic calculus for this process. We will also go through some turbulence and mathematical finance applications. The remaining of this paper is as follows. Section2 discusses the definition of FBM and Liouville form of this process. Then, some introductory material on fuzzy stochastic processes and fuzzy stochastic integrals is reviewed. Section 3 introduces a class of FFSDEs driven by FBM. Furthermore, the existence-uniqueness of solutions is proven using an approximation approach. In Section 4, some findings are presented. In section 5, application to a model of population dynamics is also illustrated. Finally, in Section 6, a conclusion is given.


## 2. Preliminaries

A Gaussian processBH=BHν is called FBM of Hurst parameter H∈0,1 if it has mean zero and the covariance function:
(3)RBHν,s=EBHνBHs=12s2H+ν2H−ν−s2H.This phenomenon was first described in [24] and investigated in [25], where a Brownian motion-based stochastic integral representation was constructed. For H>3/2, this process’ long-range dependence and self-similarity qualities give suitable driving noise in stochastic models like networks, finance, and physics. BecauseBHis not semimartingale ifH≠3/2. In terms of FBM, classical Ito^ theory cannot be used to generate stochastic integral. Two approaches were used to define stochastic integrals about FBM. In the situation of H>3/2, Young’s integral [26] can be used to define the Riemann-Stieltjes stochastic integral. The Malliavin calculus is used in a second way to define stochastic integral concerning FBM (see [27–30]). The following is an illustration of BHν given in [25]:
(4)BHν=1Γ1+α∫−∞0ν−sα−−sαdWs+BHν,where W is a Browian motion, α=H−3/2 and BHν=∫0νν−sαdWs. A FBM in Liouville form (LFBM) is the process BHν with H∈1,2, which has many of the same qualities as the FBM except for the non-stationary increments. In [27], a semimartingale process was utilized to approximate BHν using the Malliavin calculus technique:
(5)BH,εν=∫0νν−s+εαdWs,ε>0red.Furthermore,(6)BH,εν=α∫0νψεsds+εαWν,where
(7)ψεν=∫0νν−s+εα−1dWs.The processBH,εν converges to BHν in L2Ω when ε tends to zero [31].Preliminaries on FRVs, fuzzy stochastic processes (FSP), and fuzzy stochastic integrals are provided in this section (see [4, 29, 32]). The family of all nonempty, compact, and convex subsets of Rm is denoted by GRm. The Hausdorff metric, abbreviated as dH, is defined as follows:
(8)dHN,B=maxsupn∈Ninfb∈Kn−b,supb∈Binfn∈Nn−bred.With regard todH, the space GRm is a full and separable metric space. If N,B, and Q are equal to GRm, then
(9)dHN+Q,B+℘=dHN,B.A probability space is definedΩ,A,℘. If mapping F:Ω⟶GRm satisfies the following conditions, it is called N-measurable.
(10)ω∈Ω:Fω∩Q≠φ∈N.Q⊂Rm for every closed set, let MΩ,N;GRm denote a set of A-measurable multifunctions with GRm values. For ℘≥1, a multifunction F∈M is said to be L℘-integrably bounded if h∈L℘Ω,N,℘;Rm+ exists that is F≤h℘−a.e,Rm+=0,∞, and
(11)F=dHF,0=supf∈Ff.F∈M is known to be L℘-integrably bounded if and only if F∈L℘Ω,N,℘;Rm+ (see [31]). Let us put it this way:
(12)L℘Ω,N,℘;GR=F∈MΩ,N;GR:F∈L℘Ω,N,℘;Rm+.For fuzzy setu∈Rm, membership function u:Rm⟶1,2 is defined, where ux denotes degree of membership of x in fuzzy set u. Assume fuzzy sets u:Rm⟶1,2 denoted by FRm that is uα∈GRm for every α∈1,2, where uα=x∈Rm:ux≥α. Define dα:FRm×FRm⟶0,∞ by
(13)d∞u,v=sup1≤α≤2dHvα,vαred.Therefore, inFRm, d∞ is metric, and FRm,d∞ is complete metric space. We have below properties for any u,v,w,z∈FRm,λ∈Rm:
(i)
d∞u+w,v+w=d∞u+v(ii)
d∞u+w,v+z=d∞u+w+d∞w,v(iii)
d∞≤d∞u,w+d∞w,v(iv)
d∞λu,λv=λd∞u,vWe use1∈FRm as 1=21, where for y∈Rm,2yx=2 if x=y and 2yx=1 if x≠y.Definition 1 See ([33]).
A probability space is defined asΩ,N,℘. If mapping Xα:Ω⟶GRm is an N-measurable multifunction for all α∈1,2, FRV is function X:Ω⟶GRm.
Assume metricρ in set FRm and η-algebra Bρ, which is created by topology induced by N fuzzy random variable (FRV) can be thought of as measurable mapping between two measurable spaces Ω,N and FRm,Bρ, which we refer to as X is NBρ-measurable. Take a look at the below metric:
(14)dsu,v=maxinfλ∈∧supν∈1,2λν,ν,supν∈1,2dHXuν,Xuλν,where ∧ represents set of strictly increasing continuous functions: λ:1,2⟶1,2, where λ1=1,λ2=2, and Xu,Xv:1,2⟶FRm are cÃ dlÃ g representations for fuzzy sets u,v∈FRm (see [34]). The space FRm,ds is a Polish metric space, and space FRm,d∞ is complete and non-separable.
We have
â€“X is FRV if and only if X is NBds-measurable for mapping X:⟶FRm on probability space Ω,N,℘.IfX is NBds-measurable, it is FRV, but not the other way.Definition 2.
IfXα∈L℘Ω,N,℘;GRm, for any α∈1,2, FRV X:Ω⟶FRm is L℘-integrably bounded for ℘≥2.
Let us denote byL℘Ω,N,℘;FRm set of all L℘-integrably bounded FRVs. The random variables X,Y∈L℘Ω,N,℘;FRm are identical if ℘d∞X,Y=1=2. For FRV X:Ω⟶FRm, and ℘≥1, the below conditions are equivalent:
(15)X∈L℘Ω,N,℘;FRm,X0∈L℘Ω,N,℘;GRm,X0∈L℘Ω,N,℘;Rm+red.
Assumej≔0,I, and Ω,N,℘ be complete probability space with filtration Nνν∈j satisfying hypotheses, an increasing and right continuous family of sub ρ-algebras of N, and containing all ℘-null sets.Definition 3.
If mappingXν:Ω⟶FRm, for every ν∈j, is FRV, then X:j×Ω⟶FRm is FSP.Definition 4.
A FSPXisd∞-continuous, if almost all its trajectories, that is mappings X.,ω:j×Ω⟶FRm are d∞-continuous functions.
A FSPX is measurable, if Xα:jΩ⟶GRm is Bj⊗N-measurable multifunction for all α∈1,2, where Bj denotes Borel ρ-algebra of subsets of j.A processX is nonanticipating if and only if for every α∈1,2, multifunction Xα is measurable with respect to ρ-algebra A, where it is defined as follows:
(16)A≔N∈K⊗N:Nν∈Nνforeveryν∈j,where Nν=ω:ν,ω∈N.Definition 5.
A FSPX is called L℘-integrably bounded ℘≥2, if there exists real-valued stochastic process h∈L℘jΩ,A;FRm, the Fubini theorem, fuzzy integral is defined by
(17)∫0IXs,ωds,for ω∈Ω\Ax, where Ax∈N and PAx=1. The fuzzy integral ∫0IXs,wds can be defined level-wise. For every α∈1,2, and ω∈Ω\Ax, Aumann integral ∫0IXs,ωαds belongs to GRm, so FRV ∫0IXs,ω belongs to FRm, so FRV ∫0IXs,ωds belongs to FRm∀ω∈Ω\Ax.Definition 6.
The fuzzy stochastic Lebesgue-Aumann integral ofX∈L1j×Ω,A;FRm is defined as
(18)Lxν,ω=∫0I20,νsXs,ωdsforeveryω∈Ω\Ax,0foreveryω∈Ax.Proposition 7 See ([5]).
The properties of the integralLx can be demonstrated as follows:
Suppose℘≥2. If X∈L℘j×Ω,A;FRm, then Lx.,.∈L℘j×Ω,A;FRm.
(i)
SupposeX∈L1j×Ω,A;FRm, then Lxνν∈j is d∞-continuous(ii)
SupposeX,Y∈L℘j×Ω,A;FRm, for ℘≥1, then(iii)
supu∈0,νd∞℘Lν,xu,Lν,yu≤νp−1∫0νd∞pXs,Ysds.
Let us define an embedding ofRm into FRm by .:Rm⟶FRm, which is for r′∈Rm,
(19)k′a=1forn=k′,0forn∈Rm\k′.
IfX:Ω⟶Rm is random variable on probability space Ω,N,℘, now X:Ω⟶FRm is FRV. The same property exists for stochastic processes.
We define fuzzy stochastic Ito^ integral by using FRV as ∫0IXsdWs, where W is Wiener process. [5] will be beneficial for the following properties.Proposition 8.
AssumeX∈L2j×Ω,A;Rm, then ∫0νXsdWsν∈j is FSP, and we have ∫0νXsdWs∈L2j×Ω,A;FRm.Proposition 9.
SupposeX∈L2j×Ω,A;Rm, then ∫0νXsdWsν∈jisd∞-continuous.


## 3. Application to Fuzzy Stochastic Differential Equation

The following is a list of FFSDEs driven by FBM that we will investigate in this section:(20)Xν=QqνX0−mX+GqνX1+1γ∫0νν−sq−1Pqν−sfs,Xs,DβXsds+∫0νν−sq−1Pqν−sgs,Xs,DβXsdBHs,such that
(21)Qqν=MqθCtqθdθ,Gqν=∫0νCqsds,Pqν=∫0∞qθMqθCtqθdθ,where Qqν and Gqν are continuous with C0=I and K0=I, Qqν≤c,c>1 and Gqν≤c,c>1,∀ν∈0,T. Equation (20) shows that the Liouville form of FBM with H∈1,2,X0:Ω⟶FRm is an FRV, g:j×Ω×FRm⟶Rm. The approximate corresponding equation (20) is
(22)Xεν=QqνX0−mX+GqνX1+1γ∫0νν−sq−1Pqν−sfs,Xεs,DβXεsds+∫0νν−sq−1Pqν−sgs,Xεs,DβXεsdBHεs.Assumption 10.
Consider the below assumptions about coefficients of equation:
A1 The mappings f:j×Ω×FRm⟶FRm and g:j×Ω×FRm⟶Rm are N⊗AdsBds-measurable and A⊗BdsBRm-measurable, respectively
A2 There exists constant L>0∀u,v∈FRm and every ν∈j such that
(23)maxd∞fν,ω,u,fν,ω,v,gν,ω,u−gν,ω,v≤Ld∞u,v
A3 There exists constant Q>0∀u,v∈FRm and every ν∈j,
(24)maxd∞fν,ω,u,0,gν,ω,u≤Q1+d∞u,0Proposition 11. See [5].
SupposeX,Y∈L2j×Ω,A;Rm, then
(25)Esupu∈0,νd∞2∫0uXsdWs,∫0uYsdWs≤4E∫0νd∞2Xs,Ysds,for every ν∈j.Theorem 12.
Assumef:j×Ω×FRm⟶FRm and g:j×Ω×FRm⟶Rm as mappings satisfy assumptions A1−A3 and X0∈L2Ω,N0,℘;FRm. Then, Equation (22) has a strong unique solution.Proof.
Assume SDE (22),
(26)Xεν=QqνX0−mX+GqνX1+∫0νν−sq−1Pqν−sfs,Xεs,DβXεsds+∫0νν−sq−1Pqν−sgs,Xεs,DβXεsdBHεs.
By Equation (6), we can write
(27)Xεν=QqνX0−mX+GqνX1+∫0νν−sq−1Pqν−sfs,Xεs,DβXεsds+∫0ναφεsν−sq−1Pqν−sgs,Xεs,DβXεsds+∫0νεαν−sq−1Pqν−sgs,Xεs,DβXεsdWs.Let us consider the Picard iterations(28)Xnεν=QqνX0−mX+GqνX1+∫0νν−sq−1Pqν−sfs′,Xn−1εs,DβXn−1εsds+∫0ναφεsν−sq−1Pqν−sgs,Xn−1εsDβXn−1εsds+∫0νεαν−sq−1Pqν−sgs,Xn−1εsDβXn−1εsdWs,for n =1,2,..., and for every ν∈j, and X0ν=X0. For ν∈j and n∈A we denote
(29)ℓ1ν=Esupu∈0,νd∞2∫0uν−sq−1Pqν−sfs,X0εs,DβX0εsds+∫0uαφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds+∫0uεαν−sq−1Pqν−sgs,X0εs,DβX0εsdWs,0≤3Esupu∈0,νd∞2∫0uν−sq−1Pqν−sfs,X0εs,DβX0εsds,0+3Esupu∈0,νd∞2∫0uαφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds,0+3Esupu∈0,νd∞2∫0uεαν−sq−1Pqν−sgs,X0εs,DβX0εsdWs,0≤3νE∫0νd∞2ν−sq−1Pqν−sfs,X0ε,DβX0εs,0ds+3α2Esupu∈0,νd∞2∫0uφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds,0+12ε2αE∫0νd∞2ν−sq−1Pqν−sgs,X0εs,DβX0εs,0ds≤6Q2T+4ε2α1+EXε02ν+3α2Esupu∈0,νd∞2∫0uφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds,0,for α=H−3/2>1. Hence
(30)ℓ1ν≤6Q2T+4ε2α1+EXε2ν+3α2Esupu∈0,νd∞2∫0uφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds,0.We have(31)Esupu∈0,νd∞2∫0uφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds,0≤Esupu∈0,νdH2∫0uφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds,0≤Esupu∈0,ν∫0uφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds2.By applying (7)–(31) and Holder inequality, we have
(32)Esupu∈0,ν∫0uφεsν−sq−1Pqν−sgs,X0εs,DβX0εsds2=Esupu∈0,ν∫0u∫0ss−k′+εα−1dWk′ν−sq−1Pqν−sgs,X0εs,DβX0εsds2=Esupu∈0,ν∫0u∫0uν−sq−1Pqν−sgk′,X0εk′,DβX0εk′k′−s+εα−1dk′dWs2≤4E∫0ν∫sνν−sq−1Pqν−sgk′,X0εk′,DβX0εk′k′−s+εα−1dk′2ds≤4E∫0ν∫sνν−sq−1Pqν−sg2k′,X0εk′k′−s+εα−1dk′∫sνr′−s+εα−1dr′ds≤4α2ν+εαE∫0νν−sq−1Pqν−sg2k′,X0εk′,DβX0εk′k′+εαdk′≤4α2ν+ε2αE∫0νν−sq−1Pqν−sg2k′,X0εk′,DβX0εk′dk′≤8Q2α2T+ε2α1+EX0ε2ν.Hence, from (30) and (32), we obtain
(33)j1ν≤6Q2T+4ε2α+4T+ε2α1+EX0ε02νν,for every ν∈I. Then, similarly,
(34)ℓn+1ν≤3ν+4ε2α+4ν+ε2αL2E∫0νd∞2Xnεu,Xn−1εuds≤3ν+4ε2α+4ν+ε2αL2∫0νEsupu∈0,sd∞2Xnεu,Xn−1εuds≤3ν+4ε2α+4ν+ε2αL2∫0νℓnsds.Therefore,(35)ℓnν≤2Q23nν+4ε2α+4ν+ε2αn1+EX0ε02L2n−1tnn!,ν∈j,n∈N.Using Chebyshev inequality, we can determine(36)Psupu∈Id∞2Xnεu,Xn−1εu>12n≤2nℓnI.The series∑n=1∞2nℓnT is convergent. Using Borel-Cantelli lemma, we have
(37)Psupu∈Id∞2Xnεu,Xn−1εu>12ninfinitelyoften=0.There is an0ω such that for approximately every ω∈Ω,
(38)supu∈Id∞2Xnεu,∞2Xn−1εu≤12n,ifn≥n0.The sequenceXnε.,ω is uniformly convergent to d∞2-continuous fuzzy process Xε.,ω for every ω∈Ωε, in which Ωc∈N and ℘Ωc=2. We can define the mapping Xε:j×Ω⟶FRm, as Xε.,ω=Xε.,ω if ω∈Ωc and Xε.,ω as freely chosen fuzzy function when ω∈ΩΩc. For every α∈1,2 and every ν∈j with a.e., we have
(39)dHXnενα,Xενα⟶0asn⟶∞.Hence,Xε will be continuous FSP. Therefore, by Xnε∈L2j×Ω,A;FRm, we have Xε∈L2j×Ω,A;FRm. Consequently, as n approaches infinity, we can prove
(40)Esupν∈Id∞2Xnεν,Xεν+d∞2Xnεν,QqνX0ε+GqνX1ε+∫0νν−sq−1Pqν−sfs,Xεs,DβXεsds+∫0νν−sq−1Pqν−sgs,Xεs,DβXεsdBHεs2,tends to zero. Now,
(41)Esupν∈Id∞2Xεν,QqνX0ε+GqνX1ε+∫0νν−sq−1Pqν−sfs,Xεs,DβXεsds+∫0νν−sq−1Pqν−sgs,Xεs,DβXεsdBHεs=0.Therefore,(42)supν∈Id∞2Xεν,QqνX0ε+GqνX1ε+∫0νν−sq−1Pqν−sfs,Xεs,DβXεsds+∫0νν−sq−1Pqν−sgs,Xεs,DβXεsdBHεs=0.This demonstrates existence of a strong solution.We suppose thatXε,Yε:j×Ω⟶FRm are strong solutions. Let
(43)ℓν=Esupu∈0,Td∞2Xεu,Yεu.Then, using computations similar to those used in existing case, we obtain(44)ℓν≤3ν+4ε2α+4ν+ε2αL2E∫0νd∞2Xεs,Yεsds≤3T+4ε2α+α2L2∫0νℓsds.When the Gronwall inequality is applied,ℓν=0 is obtained for ν∈j. Therefore,
(45)supν∈jd∞2Xεν,Yεν=0,which completes the uniqueness proof.Lemma 13.
For everyε>0 and 1<α<3/2, we have
(46)∫sνk′−s+εα−1−k′−sα−1dk′≤α+1αεα.Proof.
To obtain the functionfx=xα−1, we use the finite-increments formula:
(47)x+εα−1−xα−1=α−1x+θεα−2,1<θ<2,then
(48)k′−s+εα−1−k′−sα−1≤α−1k′−sα−2ε.
Hence,(49)∫sνk′−s+εα−1−k′−sα−1dk′=∫ss+εk′−s+εα−1−k′−sα−1dk′+∫s+ενk′−s+εα−1−k′−sα−1dk′≤∫ss+ε2k′−sα−1−k′−sα−1dk′+α−1ε∫s+ενk′−sα−2dk′.
Therefore,(50)∫sνk′−s+εα−1−k′−sα−1dk′≤∫ss+εk′−sα−1dk′+α−1ε∫s+ενk′−sα−2dk′≤1αεα+α−1ε1α−1εα−1=α+1αεα.Proposition 14.
InL2j×Ω, the solution Xεν of Equation (22) converges to solution Xν of Equation (20) as ε⟶0 uniformly with regard to ν∈0,I.Proof.
Assume the following approximation of equations based on Equation (20):
(51)Xν=QqνX0−mX+GqνX1+∫0νν−sq−1Pqν−sfs,Xs,DβXsds+∫0νν−sq−1Pqν−sgs,Xs,DβXsdBHs,Xεν=QqνX0−mX+GqνX1+∫0νν−sq−1Pqν−sfs,Xεs,DβXεsds+∫0νν−sq−1Pqν−sgs,Xεs,DβXεsdBHεs.We can write(52)Esupu∈0,νd∞2Xu,Xεu≤2Esupu∈0,ν∫0ud∞2fs,Xs,DβXs,fs,Xεs,DβXεsds+2Esupu∈0,νd∞2∫0∞ν−sq−1Pqν−sgs,Xs,DβXsdBHs,∫0uν−sq−1Pqν−sgs,Xεs,DβXεsdBHεs.Then(53)Esupu∈0,νd∞2Xu,Xεu≤2Esupu∈0,ν∫0ud∞2fs,Xs,DβXs,fs,Xεs,DβXεsds+4Esupu∈0,νd∞2∫0uν−sq−1Pqν−sgs,Xs,DβXεsdBHs,∫0uν−sq−1Pqν−sgs,Xs,DβXεsdBHεs+4Esupu∈0,νd∞2∫0uν−sq−1Pqν−sgs,Xs,DβXεsdBHεs,∫0uν−sq−1Pqν−sgs,Xεs,DβXεsdBHεs=2E∫0νd∞2fs,Xs,DβXs,fs,Xεs,DβXεsds+4Esupu∈0,I∫0νν−sq−1Pqν−sgs,XεsDβXεsdBHεs−dBHs2+4Esupu∈0,ν∫0νgs,XsDβXs−gs,XεsDβXεsdBHεs2.To get the solution, consider Equation (6):
(54)Esupu∈0,νd∞2Xu,Xεu≤2E∫0νd∞2fs,Xs,DβXs,fs,Xεs,DβXεsds+8ε2αEsupu∈0,ν∫0uν−sq−1Pqν−sgs,Xεs,DβXεsdWs2+8α2Esupu∈0,ν∫0uν−sq−1Pqν−sgs,Xεs,DβXεs∫0ss−k′+εα−1−s−k′α−1dWk′ds2+8α2Esupu∈0,ν∫0ugs,Xs,DβXεs−gs,Xεs∫0ss−k′+εα−1dWk′ds2+8ε2αEsupu∈0,ν∫0ugs,Xs,DβXs−gs,Xεs,DβXεsdWs2.Then,(55)Esupu∈0,νd∞2Xu,Xεu≤2E∫0νd∞2fs,Xs,DβXs,fs,Xεs,DβXεsds+8ε2αE,supu∈0,ν∫0uν−sq−1Pqν−sgs,Xεs,DβXεsdWs2+8α2E,supu∈0,ν∫0u∫suν−sq−1Pqν−sgk′,Xεk′,DβXk′εk′−s+εα−1−s−k′α−1dk′dWs2+8α2Esupu∈0,ν∫0ugk′,Xk′−gs,Xεs,DβXsεk′−s+εα−1dk′dWs2+8ε2αEsupu∈0,ν∫0ugs,Xs,DβXs−gs,Xεs,DβXεsdWs2.Apply the Holder inequality, Doob inequality, and Ito^ isometry property to gain
(56)Esupu∈0,νd∞2Xu,Xεu≤2E∫0νd∞2fs,Xs,DβXs,fs,Xεs,DβXεsds+32ε2α∫0νν−sq−1Pqν−sg2s,Xεs,DβXsds+32α2E∫0ν∫sνν−sq−1Pqν−sg2k′,Xεk′,DβXεk′k′−s+εα−1−k′−sα−1dk′×∫sνk′−s+εα−1−k′−sα−1dk′ds+32α2E∫0ν∫sνgk′,Xk′−gs,Xεs,DβXεs2k′−s+εα−1dk′×∫sνk′−s+εα−1dk′ds+32ε2αE∫0νgs,Xs,DβXs−gs,Xεs,DβXεs2ds.We conclude that using similar arguments to (32), from (46), and assumptions A1−A3:
(57)Esupu∈0,νd∞2Xu,Xεu≤2L2E∫0νd∞2Xs,Xεsds+64ε2αQ2∫0ν1+EXεs0ds+64Q2T+ε2αα+1αεα∫0ν1+EXεs02ds+32L2T+ε2αE∫0νd∞2Xs,Xεsds+32L2ε2αE∫0νd∞2Xs,Xεsds.Hence,(58)Esupu∈0,νd∞2Xu,Xεu≤2L2+32L2T+ε2α+32L2ε2α∫0νEsupu∈0,sd∞2Xu,Xεuds+64ε2αQ2+64Q2T+ε2αα+1αεα∫0ν1+EXεs22ds.The proof is completed by applying Gronwall’s lemma toEsupu∈0,νd∞2Xu,Xεu⟶0 as ε⟶0.


### 3.1. Example in Finance

The preceding crisp SFDE is often used in financial modeling:(59)Xν=QqνX0−mX+GqνX1+∫0νμν−sq−1Pqν−sXsds+∫0νσν−sq−1Pqν−sXνdBHs,where FBM is underlying stochastic process. Because of FBM’s long-range dependence and self-similarity property, it is a good fit for describing financial quantities. On the other hand, the equation that includes the uncertainties can be used to model the price dynamics. As a result, fuzzy processes in equations are used to model. When dealing with linear coefficients, we get an explicit solution to Equation (20). Suppose FSDE that meet Theorem 12’s assumptions as follows:
(60)Xν=QqνX0−mX+GqνX1+∫0νμν−sq−1Pqν−sXsds+∫0νσ2ν−sq−1Pqν−sXl1s+Xu1sdBHs,X:Rm+×Ω⟶FRm,BH is an FBM, Xl1,Xu1:Rm+×Ω⟶Rm that is Xν1=Xl1ν,Xu1ν,X0∈L2Ω,N0,℘;FRm, and μ,σ∈Rm. To find the closed explicit form of the solution to μ≥0, we need to solve below system of equations (60).
(61)Xl1ν=QqνXl10+GqνXl′10+∫0νμν−sq−1Pqν−sXl1sds+∫0νσ2ν−sq−1Pqν−sXl1s+Xu1sdBHs,Xu1ν=QqνXu10+GqνXu′10+∫0νμν−sq−1Pqν−sXu1sds+∫0νσ2ν−sq−1Pqν−sXl1s+Xu1sdBHs,then,
(62)Xl1ν+Xu1ν=QqνXl10+GqνXl′10+QqνXu10+GqνXu′10+∫0νμν−sq−1Pqν−sXl1s+Xu1ds+∫0νσν−sq−1Pqν−sXl1s+Xu1sdBHs.The approximation solution of Equation (62) given by Equation (6) is
(63)Xlε1ν+Xuε1ν=QqνXlε10+GqνXl′ε10+QqνXuε10+GqνXu′ε10+∫0νμ+σαφεsν−sq−1Pqν−sXlε1s+Xuε1ds+∫0νσεαν−sq−1Pqν−sXlε1s+Xuε1sdWs.Therefore, one can derive a unique solution from explicit solution of crisp linear SDEs.(64)Xlε1ν+Xuε1ν=QqνXlε10+GqνXl′ε10+QqνXuε10+GqνXu′ε10expμν+σα∫0νφεsds−12σ2ε2αν+σεαWν=QqνXlε10+mXlε10+GqνXl′ε10+QqνXuε10+GqνXu′ε10expμν+σBHεν−12σ2ε2αν.Now, we apply a similar method to eachα∈1,2 to generate the following system:
(65)Xlεαν=QqνXlεα0+GqνXl′εα0+∫0νμν−sq−1Pqν−sXlεαsds+∫0νσ2ν−sq−1Pqν−sXlε1s+Xuε1sdBHεs,Xuεαν=QqνXlεα0+GqνXl′εα0+∫0νμν−sq−1Pqν−sXuεαsds+∫0νσ2ν−sq−1Pqν−sXlε1s+Xuε1sdBHεs.We use solution (64) to get below system for μ≥0:
(66)Xlεαν=QqνXlεα0+GqνXl′εα0+∫0νμν−sq−1Pqν−sXlεαsds+∫0νσ2ν−sq−1Pqν−sXlε10+Xuε10expμs+σBHεs−12σ2ε2αsdBHεs,Xuεαν=QqνXuεα0+GqνXu′εα0+∫0νμν−sq−1Pqν−sXuεαsds+∫0νσ2ν−sq−1Pqν−sXlε10+Xuε10expμs+σBHεs−12σ2ε2αsdBHεs.Alternatively, we haveW in terms of the Wiener process:
(67)Xlεαν=QqνXlεα0+GqνXlεα0+∫0νμXlεαs+ασ2Xlε10+Xuε1×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+εασ2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs,(68)Xuεαν=QqνXuεα0+GqνXuεα0+∫0νμXuεαs+ασ2Xlε10+Xuε1×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+εασ2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs.Use Theorem12 in [35] to find unique solution to (67) in the following format:
(69)Xlεαν=eμνQqνXlεα0+eμνGqνXlεα0+eμνασ2Xlε10+Xuε1×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+eμνεασ2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs,Xuεαν=eμνQqνXlεα0+eμνGqνXlεα0+eμνασ2Xlε10+Xuε1×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+eμνεασ2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs.Then,(70)Xlεαν=eμνQqνXlεα0+GqνXl′εα0+σ2Xlε10+Xuε10×∫0νexpσBHεs−12σ2ε2αsdBHεs,Xuεαν=eμνQqνXuεα0+GqνXu′εα0+σ2Xlε10+Xuε10×∫0νexpσBHεs−12σ2ε2αsdBHεs.As a result, the fuzzy approximation solution forμ≥0 is
(71)Xεν=QqνeμνX0+GqνeμνX′0+σ2Xlε10+Xuμν0eμν∫0νexpσBHεs−12σ2ε2αsdBHεs.Forμ<0, we can demonstrate that
(72)Xlεαν=QqνXlεα0+GqνXl′εα0+∫0νμν−sq−1Pqν−sXuεαsds+ασ2Xlε10+Xuε10×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+εαα2Xlε10+Xuε10×,(73)Xuεαν=QqνXuεα0+GqνXu′εα0+∫0νμν−sq−1Pqν−sXlεαsds+ασ2Xlε10+Xuε10×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+εαα2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs.The following matrix represents the unique solution to (72):
(74)Xlεαν=QqνXlεα0coshμν+GqνXl′εα0coshμν+QqνXuεα0sinhμν+GqνXu′εα0sinhμν+eμνασ2Xlε10+Xuε10×∫0νφεsexpσα∫0sφεudu−12σ2ε2αs+σεαWsds+eμνεαα2Xlε10+Xuε10×∫0νexpσα∫0sφεudu−12σ2ε2αs+σεαWsdWs,Xuεαν=QqνXlεα0coshμν+GqνXl′εα0coshμν+QqνXuεα0sinhμν+GqνXu′εα0sinhμν+eμνασ2Xlε10+Xuε10×∫0νφεsexpσα∫0sφεudu−12σ2ε2αs+σεαWsds+eμνεαα2Xlε10+Xuε10×∫0νexpσα∫0sφεudu−12σ2ε2αs+σεαWsdWs.Then,(75)Xlεαν=QqνXlεα0coshμν+GqνXl′εα0coshμν+QqνXuεα0sinhμν+GqνXu′εα0sinhμν+eμνσ2Xlε10+Xuε10∫0νexpσBHεs−12σ2ε2αsdBHεs,Xuεαν=QqνXlεα0coshμν+GqνXl′εα0coshμν+QqνXuεα0sinhμν+GqνXu′εα0sinhμν+eμνσ2Xlε10+Xuε10∫0νexpσBHεs−12σ2ε2αsdBHεs.As a result, the fuzzy approximation solution forμ<0 is
(76)Xεν=QqνX0coshμν+GqνX′0coshμν+QqνX0sinhμν+GqνX′0sinhμν+σ2Xlε10+Xuε10eμν∫0νexpσBHεs−12σ2ε2αsdBHεs.


## 3.1. Example in Finance

The preceding crisp SFDE is often used in financial modeling:(59)Xν=QqνX0−mX+GqνX1+∫0νμν−sq−1Pqν−sXsds+∫0νσν−sq−1Pqν−sXνdBHs,where FBM is underlying stochastic process. Because of FBM’s long-range dependence and self-similarity property, it is a good fit for describing financial quantities. On the other hand, the equation that includes the uncertainties can be used to model the price dynamics. As a result, fuzzy processes in equations are used to model. When dealing with linear coefficients, we get an explicit solution to Equation (20). Suppose FSDE that meet Theorem 12’s assumptions as follows:
(60)Xν=QqνX0−mX+GqνX1+∫0νμν−sq−1Pqν−sXsds+∫0νσ2ν−sq−1Pqν−sXl1s+Xu1sdBHs,X:Rm+×Ω⟶FRm,BH is an FBM, Xl1,Xu1:Rm+×Ω⟶Rm that is Xν1=Xl1ν,Xu1ν,X0∈L2Ω,N0,℘;FRm, and μ,σ∈Rm. To find the closed explicit form of the solution to μ≥0, we need to solve below system of equations (60).
(61)Xl1ν=QqνXl10+GqνXl′10+∫0νμν−sq−1Pqν−sXl1sds+∫0νσ2ν−sq−1Pqν−sXl1s+Xu1sdBHs,Xu1ν=QqνXu10+GqνXu′10+∫0νμν−sq−1Pqν−sXu1sds+∫0νσ2ν−sq−1Pqν−sXl1s+Xu1sdBHs,then,
(62)Xl1ν+Xu1ν=QqνXl10+GqνXl′10+QqνXu10+GqνXu′10+∫0νμν−sq−1Pqν−sXl1s+Xu1ds+∫0νσν−sq−1Pqν−sXl1s+Xu1sdBHs.The approximation solution of Equation (62) given by Equation (6) is
(63)Xlε1ν+Xuε1ν=QqνXlε10+GqνXl′ε10+QqνXuε10+GqνXu′ε10+∫0νμ+σαφεsν−sq−1Pqν−sXlε1s+Xuε1ds+∫0νσεαν−sq−1Pqν−sXlε1s+Xuε1sdWs.Therefore, one can derive a unique solution from explicit solution of crisp linear SDEs.(64)Xlε1ν+Xuε1ν=QqνXlε10+GqνXl′ε10+QqνXuε10+GqνXu′ε10expμν+σα∫0νφεsds−12σ2ε2αν+σεαWν=QqνXlε10+mXlε10+GqνXl′ε10+QqνXuε10+GqνXu′ε10expμν+σBHεν−12σ2ε2αν.Now, we apply a similar method to eachα∈1,2 to generate the following system:
(65)Xlεαν=QqνXlεα0+GqνXl′εα0+∫0νμν−sq−1Pqν−sXlεαsds+∫0νσ2ν−sq−1Pqν−sXlε1s+Xuε1sdBHεs,Xuεαν=QqνXlεα0+GqνXl′εα0+∫0νμν−sq−1Pqν−sXuεαsds+∫0νσ2ν−sq−1Pqν−sXlε1s+Xuε1sdBHεs.We use solution (64) to get below system for μ≥0:
(66)Xlεαν=QqνXlεα0+GqνXl′εα0+∫0νμν−sq−1Pqν−sXlεαsds+∫0νσ2ν−sq−1Pqν−sXlε10+Xuε10expμs+σBHεs−12σ2ε2αsdBHεs,Xuεαν=QqνXuεα0+GqνXu′εα0+∫0νμν−sq−1Pqν−sXuεαsds+∫0νσ2ν−sq−1Pqν−sXlε10+Xuε10expμs+σBHεs−12σ2ε2αsdBHεs.Alternatively, we haveW in terms of the Wiener process:
(67)Xlεαν=QqνXlεα0+GqνXlεα0+∫0νμXlεαs+ασ2Xlε10+Xuε1×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+εασ2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs,(68)Xuεαν=QqνXuεα0+GqνXuεα0+∫0νμXuεαs+ασ2Xlε10+Xuε1×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+εασ2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs.Use Theorem12 in [35] to find unique solution to (67) in the following format:
(69)Xlεαν=eμνQqνXlεα0+eμνGqνXlεα0+eμνασ2Xlε10+Xuε1×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+eμνεασ2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs,Xuεαν=eμνQqνXlεα0+eμνGqνXlεα0+eμνασ2Xlε10+Xuε1×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+eμνεασ2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs.Then,(70)Xlεαν=eμνQqνXlεα0+GqνXl′εα0+σ2Xlε10+Xuε10×∫0νexpσBHεs−12σ2ε2αsdBHεs,Xuεαν=eμνQqνXuεα0+GqνXu′εα0+σ2Xlε10+Xuε10×∫0νexpσBHεs−12σ2ε2αsdBHεs.As a result, the fuzzy approximation solution forμ≥0 is
(71)Xεν=QqνeμνX0+GqνeμνX′0+σ2Xlε10+Xuμν0eμν∫0νexpσBHεs−12σ2ε2αsdBHεs.Forμ<0, we can demonstrate that
(72)Xlεαν=QqνXlεα0+GqνXl′εα0+∫0νμν−sq−1Pqν−sXuεαsds+ασ2Xlε10+Xuε10×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+εαα2Xlε10+Xuε10×,(73)Xuεαν=QqνXuεα0+GqνXu′εα0+∫0νμν−sq−1Pqν−sXlεαsds+ασ2Xlε10+Xuε10×∫0νφεsexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsds+εαα2Xlε10+Xuε10×∫0νexpμs+σα∫0sφεudu−12σ2ε2αs+σεαWsdWs.The following matrix represents the unique solution to (72):
(74)Xlεαν=QqνXlεα0coshμν+GqνXl′εα0coshμν+QqνXuεα0sinhμν+GqνXu′εα0sinhμν+eμνασ2Xlε10+Xuε10×∫0νφεsexpσα∫0sφεudu−12σ2ε2αs+σεαWsds+eμνεαα2Xlε10+Xuε10×∫0νexpσα∫0sφεudu−12σ2ε2αs+σεαWsdWs,Xuεαν=QqνXlεα0coshμν+GqνXl′εα0coshμν+QqνXuεα0sinhμν+GqνXu′εα0sinhμν+eμνασ2Xlε10+Xuε10×∫0νφεsexpσα∫0sφεudu−12σ2ε2αs+σεαWsds+eμνεαα2Xlε10+Xuε10×∫0νexpσα∫0sφεudu−12σ2ε2αs+σεαWsdWs.Then,(75)Xlεαν=QqνXlεα0coshμν+GqνXl′εα0coshμν+QqνXuεα0sinhμν+GqνXu′εα0sinhμν+eμνσ2Xlε10+Xuε10∫0νexpσBHεs−12σ2ε2αsdBHεs,Xuεαν=QqνXlεα0coshμν+GqνXl′εα0coshμν+QqνXuεα0sinhμν+GqνXu′εα0sinhμν+eμνσ2Xlε10+Xuε10∫0νexpσBHεs−12σ2ε2αsdBHεs.As a result, the fuzzy approximation solution forμ<0 is
(76)Xεν=QqνX0coshμν+GqνX′0coshμν+QqνX0sinhμν+GqνX′0sinhμν+σ2Xlε10+Xuε10eμν∫0νexpσBHεs−12σ2ε2αsdBHεs.


## 4. Application to a Model of Population Dynamics

Consider a population of a particular species that lives on a specific piece of land. The number of individuals in the underlying population at the instantν is denoted by Xν. The Mslthus differential equation describes a conventional, crisp, deterministic model of population evolution:
(77)0cDναXν=N−LXν+fν,Xν,DβXνds,ν∈0,I,(78)X0+mX=X0,(79)X′0=X1.The reproduction and mortality coefficients are defined by the constantsN and L, respectively. The notation X0−mX and X1 denotes the initial number of people. The X result of this equation is
(80)Xν=X0expaν,wherea=N−L.Assume thata≠0 is true. Let us observe that the mild solution of Equation (77):
(81)Xν=QqνX0−mX+GqνX1+1γa∫0νν−sq−1Pqν−sfν,Xν,DβXνds.In the next section, we will apply the previous model to the situation when there are certain uncertainties inXν. Let us introduce an observer (who keeps an eye on this population). Assume that the population’s state is determined by random factors and that the observer can only explain the population’s state using linguistic terms, such as “very small,” “small,” “not big,” and “large”. In this way, the population growth model incorporates two types of uncertainty. The first type of uncertainty belongs to probability theory, whereas the second belongs to fuzzy set theory. We could write the model with uncertainties like this at this point:
(82)Xν,ω=QqνX0−mX+GqνX1+1γa∫0νν−sq−1Pqν−sfν,Xν,DβXν,ωds,where ω represents a random factor (a probability space Ω,N,℘ is taken into account; ω∈Ω, X0−mX, and X1 are a FRV; the integral has become a fuzzy integral; and the solution X is now a fuzzy stochastic process X:0,T×FR. Such problem 39 has its differential counterpart, and exemplifies the random fuzzy integral equations or, equivalently, random fuzzy differential equations (see 36).Consider that some people leave their homeland and aliens enter the population and that this occurs in a disorderly manner. Allow Brownian motion B to model the aggregated immigration process. The population dynamics may now be simulated using the following equation with uncertainties:(83)Xν,ω=QqνX0−mX+GqνX1+1γa∫0νν−sq−1Pqν−sfν,Xν,DβXν,ωds+Bν,ω.This equation can be rewritten as follows (we will not write the argumentω in the next section):
(84)Xν=QqνX0−mX+GqνX1+1γa∫0νν−sq−1Pqν−sfν,Xν,DβXνds+Bs.Alternatively, in a symbolic, differential form:(85)0cDναXν=N−LXν+N−LX′ν+fν,Xν,DβXνds+dBt,ν∈0,I,(86)X0+mX=X0,(87)X′0=X1,So we get to the SFDE of type 2.1, wheref:0,T×Ξ×FR⟶FR is given by fν,u=a.u, and g:0,T×Ω×FR⟶R is defined by gν,u=1. The coefficients of such equations satisfy the requirements A1−A3. Considering that X0:Ω⟶FR is a FRV with the property that X0−mX∈L2Ω,N,℘;FR,X1∈L2Ω,N,℘;FR and X0−mX,X1 are N-measurable, Equation (85), or equivalently Equation (84), has a unique solution.In the sequel, we shall establish the explicit solution to (84) with a≠0. To this end, let us denote the α-levels α∈1,2 of the solution X:0,T×Ω⟶FR and α-levels of initial value X0−mX:Ω⟶FR and X1:Ω⟶FR as
(88)Xνα=Lαν,UανandX0α=X0,Lα,L,X0,Uα,respectively. Obviously, Lα,Uα:0,T×Ω⟶R are the stochastic processes; also, X0,Lα−mXα,X0,Uα−mXα:Ω⟶R and X1,Lα,X1,Uα:Ω⟶R are the random variables. If the FSP X is a solution to (84), then for every ν∈0,T, the following property should hold:
(89)PXνα=να−1PανQqνX0−mXα+να−1PανGqνX1α+a∫0νν−sq−1Pqν−sfν,Xν,DβXνdsα+∫0tdBsα,∀α∈1,2=2.


## 5. Example

Assume the following FSDEs:(90)0cDνγXν=Xν+ν2+sin2X+cos2X,where ν∈1,2,fν,Xν=Yν+ν2,gν,Xν=sin2X+cos2X,3/2<β<2. It is easy to verify that f,σ satisfy the A1−A3. Define f¯X,Y as follows:
(91)∫12f¯X,Yν2+νdν=∫12f¯ν,X,Yν2+νdν.We can prove thatf¯X,Y=ν3/3+ν2/2. Similarly, g¯X,Y=1/2X+sin2X+1/4sin2X. The averaging form of (90) can be written as
(92)0cDνγYν=Yνν33+ν22+12X+sin2X+14sin2XdBνH.Asε approaches zero, the solutions Xν and Yν are equal in the sense of mean square, according to Theorem 12. As a result, the findings may be verified.


## 6. Conclusion

In order to model fuzziness in the Liouville form FBM, we created a fuzzy stochastic differential equation with a number of characteristics, such as long-range dependence and unpredictability. We applied an approximation strategy to fractional stochastic integrals and embedded the Ito^ classical integral in fuzzy set space. We used the Picard iteration method to examine the existence-uniqueness of solutions. We demonstrated that the approximate solution converges uniformly to the precise solution. We also demonstrate the existence and uniqueness of solutions to FFSDEs under the Lipschitzian coefficient. Illustrations of the application to a population dynamics model are also provided. The example is also illustrated at the end of the text. Future work may also involve generalizing other tasks, adding observability, and developing the concept that was introduced in this mission. This is a fertile field with numerous research initiatives that have the potential to produce a wide range of theories and applications.



---

*Source: 3559035-2022-10-11.xml*