# Some Properties of Numerical Solutions for Semilinear Stochastic Delay Differential Equations Driven by G-Brownian Motion

Authors: Haiyan Yuan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper is concerned with the numerical solutions of semilinear stochastic delay differential equations driven by G-Brownian motion (G-SLSDDEs). The existence and uniqueness of exact solutions of G-SLSDDEs are studied by using some inequalities and the Picard iteration scheme first. Then the numerical approximation of exponential Euler method for G-SLSDDEs is constructed, and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent, and it can reproduce the stability of the analytical solution under some restrictions. Numerical experiments are presented to confirm the theoretical results.


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## Body

## 1. Introduction

Many models in many branches of science and industry, such as economics, finance, biology, and medicine, reveal stochastic effects and are introduced as stochastic differential equations (SDEs). Some phenomena in various fields such as population dynamics [1], optimal pricing in economics [2], thermal noise [3], and spread of virus [4] show stochastic behaviors. However, since most SDEs cannot be solved explicitly, numerical approximations which are on the basis of incorporating the stochastic factor in the classical numerical approximations for DDEs have become an important tool in the study of SDEs. A lot of numerical results for the SDEs have been obtained; please see the works of Chassagneux et al. [5], Higham et al. [6, 7], Banihashemi et al. [8], Babaei et al. [9], Liu and Mao [10], and so on.The phenomenon of stiffness appears in the process of applying a certain numerical method to ODEs and SDEs. It is known that the stiffness makes standard explicit integrators useless. Nevertheless, the implicit scheme does not perform well for the step size reduction which is forced by accuracy requirements: the method tends to resolve all the oscillations in the solutions and hence leads to its numerical inefficiency. Due to the cost of computing the Jacobian and the exponential or related function of Jacobian, many works are directed at the semilinear stochastic problems and exploring the exponential integrators to approximate the semilinear stochastic problems as they can solve exactly the linear part and maintain some qualitative behaviors (including stability) of the exact solutions. We refer the readers to the works of Higham et al. [11], Hochbruck and Ostermann [12], Bouc and Pardoux [13], Maset and Zennaro [14], Pardoux [15], Altman [16], and Yuan [17].In the real world, we are often faced with two kinds of uncertainties, that is, probabilistic uncertainty and model uncertainty. Model uncertainty is due to incomplete information, vague data, imprecise probability, and so forth. Many researchers investigate the characteristics of model uncertainty in order to provide a framework for theory and applications. Hou et al. [18] developed the stability analysis for discrete-time uncertain time-delay systems governed by an infinite-state Markov chain. They derived some sufficient conditions for the exponential stability in mean square with conditioning via linear matrix inequalities and established the equivalence among asymptotical stability in mean square, stochastic stability, and exponential stability in mean square. Yi et al. [19] investigated the stabilization of a class of chaotic systems with both model uncertainty and external disturbance. They developed a new UDE-based control method by combining the dynamic feedback control method and the uncertainty and disturbance estimator- (UDE-) based control method. Peng [20] gave the notions of G-expectation and G-Brownian motion on sublinear expectation space which provide the new perspective for the stochastic calculus under model uncertainty, which has aroused great interest. Based on the fundamental theory of time-consistent G-expectation, Peng [21] introduced the so-called G-Gaussian distribution and the G-Brownian motion and used them to set up the associated Ito^ integral. Since then, many works have been carried out on the stochastic calculus with respect to the G-Brownian motion. One can see the works of Denis et al. [22], Dolinsky et al. [23], Faizullah et al. [24–26], Ullah and Faizullah [27], Fadina and Herzberg [28], Hu and Peng [29], Li et al. [30], Ren et al. [31], Yin and Ren [32], Yang and Zhu [33], and Zhang and Chen [34]. It can be found that most researches focus on linear and nonlinear SDEs, SDDEs, and NSDDEs with G-Brownian motion; there are a few numerical analysis results for semilinear SDDEs with G-Brownian motion (G-SLSDDEs). To fill this gap, we investigate the numerical solutions of G-SLSDDEs and give some results in the present paper.The remainder of the paper is organized as follows. In Section2, we introduce some basic notations, assumptions, and properties which will be used in this paper. We devote Section 3 to presenting the existence and uniqueness of the exact solution for the G-SLSDDEs. In Section 4, we are in a position to establish the exponential integrators for G-SLDDEs and we derive conclusions about the convergence and the exponential mean-square stability of the exact solution for the G-SLSDDEs. We are also successful in establishing the exponential Euler method and proving that the exponential Euler method can preserve the mean-square stability of the exact solution. In Section 5, numerical simulations are presented to demonstrate the theoretical results; and a short conclusion is given in Section 6.


## 2. Preliminaries

We recall some basic definitions and notions from [20, 21]. Let Ω be a (nonempty) basic space and let ℋ be a linear space of real-valued functions defined on Ω such that the constant C∈ℋ and if X1,X2,…,Xn∈ℋ, then φX1,X2,…,Xn∈ℋ for each φ∈ℂl.LipRn, where ℂl.LipRn is the space of linear function φ defined as follows:(1)ℂl.LipRn=φ:Rn⟶R|∃C∈R+,m∈N,s.t.φx−φy≤C1+xm+ymx−y,for x,y∈Rn. We consider that ℋ is the space of random variables.Definition 1.
(see [20]; sublinear expectation). A function E^:ℋ⟶R is called sublinear expectation if, ∀X,Y∈ℋ,C∈R, and λ≥0, it satisfies the following properties:(1)
Monotonicity: if X≥Y, then E^X≥E^Y(2)
Constantpreserving:E^C=C(3)
Subadditivity:E^X+Y≥E^X+E^YorE^X−E^Y≥E^X−Y(4)
Positivehomogeneity:E^λX=λE^X
Also, ifE^X=E^−X=0, then E^C+λX+Y=C+E^Y.
The tripleΩ,ℋ,E^ is called a sublinear expectation space. If (7) and (18) are satisfied, the aforementioned function E^:ℋ⟶R is called a nonlinear expectation and the triple Ω,ℋ,E^ is relevantly called a nonlinear expectation space. For the details of the notions of G-normal distribution, G-expectation, G-conditional expectation, and G-Brownian motion, see Chapters 2 and 3 of Peng [21].Definition 2.
(see [21]; G-normal distribution). Let Ω,ℋ,E^ be a sublinear expectation space, and X∈ℋ with(2)σ¯2=E^X2,σ¯2=−E^−X2.
Then,X is said to be G-distributed or N0;σ¯2,σ¯2-distributed, if, ∀a,b≥0, we have(3)aX+bY∼a2+b2X,where each Y∈ℋ is an independent copy of X; that is, Y∼X, and Y is independent from X.Definition 3.
(see [21]; G-Brownian motion). The sublinear expectation E^:LipΩ⟶R is called a G-expectation if the corresponding canonical process ωtt≥0 on the sublinear expectation space Ω,LipΩ,E^ is a G-Brownian motion; that is, for 0≤s<t, it satisfies the following conditions:(1)
ω0=0(2)
The incrementωt+s−ωs is independent of ωt1,ωt2,…,ωtn for each n∈Z+, and 0≤t1≤⋯≤tn≤t(3)
The incrementωt+s−ωs is N0,sσ¯2,sσ¯2−distributed
For each fixedT≥0, set ΩT=ω⋅∧T:ω∈Ω.(4)LipΩT=φωt1,ωt2,…,ωtn:t1,t2,…,tn∈0,T,φ∈ℂl.LipRn,n∈N,where LipΩt⊆LipΩT for t≤T and LipΩT=∪m=1∞LipΩm. The completion of LipΩ under the norm Xp=E^Xp1/p for p≥1 is denoted by LGpΩ and LGpΩt⊂LGpΩT⊂LGpΩ for 0≤t≤T<∞. An important proposition about conditional G-expectation E^⋅|Ωt, t∈0,T is presented as follows.Proposition 1.
(see [21]). The conditional expectation E^⋅|Ωt, t∈0,T holds for each X,Y∈LG1Ωt:(1)
IfX≥Y, then E^X|Ωt≥E^Y|Ωt(2)
E^η|Ωt=η, for each t∈0,∞ and η∈LG1Ωt(3)
E^X−E^Y≤E^X−Y|Ωt(4)
E^ηX|Ωt=η+E^X|Ωt+η−E^−X|Ωt for each bounded η∈LG1Ωt(5)
E^E^X|Ωt|Ωs=E^X|Ωt∧s; in particular, E^E^X|Ωt=E^X
ForT∈R+, a partition πT of 0,T is a finite-ordered subset π=t1,…,tN such that 0=t0<t1<⋯<tN=T. Let p≥1 be fixed. Define(5)MGp,00,T=ηt=ηt,ω=∑j=1Nξj−1ωItj−1,tjt;ξj−1∈LGpℱtj−1,tj−1<tj,j=1,2,…,N,t0=0,tN=T,N≥1,where LGpℱt=ξ∈LG1ℱt:E^ξp<∞. For ηt=∑j=0N−1ξjωItj,tj+1t∈MGp,00,T, set(6)E^Tη≔1T∫0TE^ηtdt=1T∑j=0N−1E^ξjtj+1−tj.
Then,E^T:MGp,00,T↦R forms a sublinear expectation. For each p≥1, MGp0,T denotes the completion of MGp,00,T under the norm(7)ηMGp,00,T≔1T∫0TE^ηtpdt1/p.Definition 4.
(see [20]; Ito^integral) For each η∈MGp,00,T with the form(8)ηt=∑j=0N−1ξjωItj,tj+1t,the Ito^ integral of G-Brownian motion is defined as(9)Iη=∫0Tηsdωas≔∑j=0N−1ξjωatj+1−ωatj.
The mappingI:MG2,00,T↦LG2ℱT can be continuously extended to I:MG20,T↦LG2ℱT. For each η∈MG20,T, the stochastic integral is defined by(10)∫0Tηsdωas≔Iη.Definition 5.
(see [20]; quadratic variation process). Let πtN=t0N,t1N,…,tNN, let N=1,2,…, be a sequence of partitions of 0,t, and set μπtN=max1≤i≤NtiN−ti−1N. An increasing process ωat,t≥0 with ωa0=0, defined by(11)ωat=limμπtN∑k=0N−1ωatk+1N−ωatkN2=ω2t2−2∫0tωasdωas,is called the quadratic variation process of G-Brownian motion. Furthermore, for each fixed s≥0,(12)ωat+s−ωas=ωsat,where ωst=ωs+t−ωs, t≥0, ωsat=a,ωst, and x,y=∑i=1dxiyi for x,y∈Rd.
Letωt≥0 be a 1-dimensional G-Brownian motion with Ga=1/2E^aω12=1/2σ¯2a+−σ¯2a−, where E^ω12=σ¯2,−E^−ω12=σ¯2, and 0≤σ¯≤σ¯. As for the definitions of Ito^ integral and the quadratic variation process with G-Brownian motion, we present two propositions in the following.Proposition 2.
(see [35]). For any 0≤t≤T<∞, let the quadratic variation of G-Brownian motion ωt=∫0tusds. Then, one obtains(13)σ¯2T−t≤ωT−ωt≤σ¯2T−tq.s.
Moreover,σ¯2dt≤ωt=utdt≤σ¯2dtq.s; and Ga≥1/2autq.s.Proposition 3.
(see [21]). For any 0≤t≤T<∞,(1)
E^∫0Tηtdωt=0; E^∫0Tηtdωt≤σ¯2E^∫0Tηtdt, ∀ηt∈MG10,T(2)
E^∫0Tηtdωt2=E^∫0Tηt2dωt,∀ηt∈MG20,T(3)
E^∫0Tηtpdt≤∫0TE^ηtpdt,∀ηt∈MG20,T,p≤1Lemma 1.
(see [36]).(1)
Letp≥1, a,a^∈Rd, η∈MGp0,T, and 0≤s≤t≤T. Then,(14)E^sups≤u≤t∫suηrdwa,wa^rp≤σa+a^a+a^T+σa−a^a−a^T4p⋅t−sp−1∫stE^ηtpdu.(2)
Letp≥2, η∈MGp0,T, and 0≤s≤t≤T. Then,(15)E^sups≤u≤t∫suηrdwrp≤CpE^∫stηu2dup/2≤Cpt−sp/2−1∫stE^ηupdu,where Cp is a positive constant independent of η.Lemma 2.
(see [37]). Doob martingale inequality). Let Xtt≥0 be an Rn-value G-martingale and let a,b be a bounded interval of R+; if p>1,Xt∈LGpΩ,Rn, then(16)E^supa≤t≤bXtp≤pp−1pE^Xbp.


## 3. Existence and Uniqueness Theorem

Throughout this paper, unless otherwise specified,a∧b and a∨b denote maxa,b and mina,b, respectively. Let τ>0 and BC−τ,0;Rd denote the family of all bound continuous Rd-valued functions φ defined on −τ,0 with norm φ=sup−τ≤θ≤0φθ. For p>0 and t≥0, let Ω,ℱ,ℱt≥0,P be a complete probability space with a filtration ℱt≥0 satisfying the usual conditions. Let ωt be a one-dimensional G-Brownian motion with ωt∼N0,σ¯2t,σ¯2t.In this paper, we focus on the G-SLSDDEs with the following form:(17)dyt=Ayt+ft,yt,yt−τdt+gt,yt,yt−τdωt+ht,yt,yt−τdωt,t∈0,T,yt=y0,t∈−τ,0,with initial condition(18)y0=ξ=φ:φisℱ0−measurable,BC−τ,0;Rd−value random variable,such thatφ∈MG2−τ,0;Rd,where A∈Rd×d, f,g,h:R×Rd×Rd⟶Rd, as well as f,g,h∈MG20,T;Rd, ∀T≥0, and ωt,t≥0 is the quadratic variation process of G-Brownian motion ωt,t≥0.The G-SLSDDEs (7) with initial value (18) can be written in the following equivalent form:(19)yt=y0+∫0tAyu+fu,yu,yu−τdu+∫0tgu,yu,yu−τdωu+∫0thu,yu,yu−τdωu.To ensure the existence and uniqueness of the solutions, we assume thatf, g, and h satisfy the following Lipschitz condition:(H1)Lipschitz condition: There exists a positive constant L1, for all x1,y1,x2, y2∈Rd, t≥0, such that(20)ft,x1,y1−ft,x2,y22∨gt,x1,y1−gt,x2,y22∨gt,x1,y1−gt,x2,y22≤L1x1−x22+y1−y22.According to (20), it is easy to obtain the following linear growth condition:(21)ft,x,y2≤2ft,x,y−ft,0,02+2ft,0,02≤L1x2+y2+2ft,0,02≤G11+x2+y2.Similarly,(22)gt,x,y≤G11+x2+y2,ht,x,y≤G11+x2+y2,where G1=L1∨2ft,0,02∨2gt,0,02.Theorem 1.
Assume thatf, g, and h satisfy the Lipschitz condition (20), and there is a nonnegative constant λ>0 such that(23)z,ft,z,0≤−λz2,hold for all t≥0,z∈Rd. Then there exists a unique solution y=yt,t∈0,∞ of equation (7) and the solution belongs to MG2R+;Rd. Moreover, the solution yt satisfies the condition(24)E^sup0≤s≤Tyns2≤4+8A2+16G1+32σ¯2G1+8G1σ¯4TE^y02+8G1T2+16σ¯2G1T+4G1σ¯4T2⋅e8A2+16G1+32σ¯2G1+8G1σ¯4T.Proof.
The proof is rather technical, and we shall divide the whole proof into several steps.Step 1.
Boundedness. For every T>0 and integer n≥1, define the stopping time(25)τn=T∧inft∈0,T:yt≥n.
Clearly,τn↑T almost surely. Set ynt=yt∧τn for t∈0,T. Then ynt satisfies the equation(26)ynt=y0+∫0tAyns+fs,yns,yns−τI0,τnsds+∫0tgs,yns,yns−τI0,τnsdωs+∫0ths,yns,yns−τI0,τnsdωs.
Using the elementary inequalitya+b+c+d2≤4a2+b2+c2+d2, we have(27)ynt2≤4y02+4∫0tAyns+fs,yns,yns−τI0,τnsds2+4∫0tgs,yns,yns−τI0,τnsdωs2+4∫0ths,yns,yns−τI0,τnsdωs2.
Therefore,(28)sup0≤s≤tyns2≤4y02+4sup0≤s≤t∫0tAyns+fs,yns,yns−τI0,τnsds2+4sup0≤s≤t∫0tgs,yns,yns−τI0,τnsdωs2+4sup0≤s≤t∫0ths,yns,yns−τI0,τnsdωs2.
Taking the G-expectation on both sides, it gives(29)E^sup0≤s≤tyns2≤4E^y02+sup0≤s≤t∫0tAyns+fs,yns,yns−τI0,τnsds2+4E^sup0≤s≤t∫0tgs,yns,yns−τI0,τnsdωs2+4E^sup0≤s≤t∫0ths,yns,yns−τI0,τnsdωs2.
It follows from Ho..lder’s inequality and the linear growth condition (21) that(30)E^sup0≤s≤t∫0tAyns+fs,yns,yns−τI0,τnsds2≤T∫0t2A2E^yns2+2G11+E^yns2+E^yns−τ2ds≤2G1T2+2A2+4G1∫0tE^sup−τ≤v≤synv2ds.
Together with Proposition3 and Doob martingale inequality, one can get(31)E^sup0≤s≤t∫0tgs,yns,yns−τI0,τnsdωs2≤4σ¯2E^∫0tgs,yns,yns−τ2ds≤4σ¯2G1E^∫0t1+yns2+yns−τ2ds≤4σ¯2G1T+8σ¯2G1∫0tE^sup−τ≤v≤synv2ds.
Similarly, it follows from Proposition2 and Ho..lder’s inequality that one gains(32)E^sup0≤s≤t∫0ths,yns,yns−τI0,τnsdωs2≤E^sup0≤s≤t∫0ths,yns,yns−τusds2≤E^∫0tus2ds∫0ths,yns,yns−τ2ds≤G1σ¯4T2+2G1σ¯4T∫0tE^sup−τ≤v≤synv2ds.
Substituting (30)–(32) into (29), one obtains(33)E^sup0≤s≤tyns2≤4E^y02+8G1T2+8A2+16G1∫0tE^sup−τ≤v≤synv2ds+16σ¯2G1T+32σ¯2G1∫0tE^sup−τ≤v≤synv2ds+4G1σ¯4T2+8G1σ¯4T∫0tE^sup−τ≤v≤synv2ds=4E^y02+8G1T2+16σ¯2G1T+4G1σ¯4T2+8A2+16G1+32σ¯2G1+8G1σ¯4T∫0tE^sup−τ≤v≤synv2ds≤4+8A2+16G1+32σ¯2G1+8G1σ¯4TE^y02+8G1T2+16σ¯2G1T+4G1σ¯4T2+8A2+16G1+32σ¯2G1+8G1σ¯4T∫0tE^sup0≤v≤synv2ds.
An application of the Gronwall inequality yields that(34)E^sup0≤s≤tyns2≤4+8A2+16G1+32σ¯2G1+8G1σ¯4TE^y02+8G1T2+16σ¯2G1T+4G1σ¯4T2⋅e8A2+16G1+32σ¯2G1+8G1σ¯4T.
Thus,(35)E^sup0≤s≤τnyns2≤4+8A2+16G1+32σ¯2G1+8G1σ¯4TE^y02+8G1T2+16σ¯2G1T+4G1σ¯4T2⋅e8|A|2+16G1+32σ¯2G1+8G1σ¯4T.
Finally, the required inequality (24) follows by letting n⟶∞.Step 2.
Uniqueness. Let yt and y¯t be two solutions of Equation (7). By the proof of boundedness, we know that both belong to MG2R+;Rd, and(36)yt−y¯t=∫0tAys−y¯s+fs,ys,ys−τ−fs,y¯s,y¯s−τds+∫0tgs,ys,ys−τ−gs,y¯s,y¯s−τdωs+∫0ths,ys,ys−τ−hs,y¯s,y¯s−τdωs.
In the same way as the proof of the boundedness, we have(37)E^sup0≤s≤tys−y¯s2≤8A2+16L1+32σ¯2L1+8L1σ¯4T∫0tE^sup0≤v≤syv−y¯v2ds.
An application of the Gronwall inequality yields that(38)E^sup0≤t≤Tyt−y¯t2=0.
Hence,yt=y¯t for all 0≤t≤T almost surely. The uniqueness has been proved.Step 3.
Existence: Set y0t=y0, for 0≤t≤T. Let y0n=φ, n=1,2,…, and for t∈0,T, define the Picard iterations(39)ynt=y0+∫0tAyn−1s+fs,yn−1s,yn−1s−τds+∫0tgs,yn−1s,yn−1s−τdωs+∫0ths,yn−1s,yn−1s−τdωs.
Obviously,y0t∈MG20,T;Rd. Moreover, it is easy to see by induction that ynt∈MG20,T;Rd; in fact,(40)ynt2≤4Ey02+4∫0tAyn−1s+fs,yn−1s,yn−1s−τds2+4∫0tgs,yn−1s,yn−1s−τdωs2+4∫0ths,yn−1s,yn−1s−τdωs2.
Taking the G-expectation on both sides, it follows from Propositions2 and 3 and Doob martingale inequality (taking p=2) that(41)E^ynt2≤4E^y02+4tE^∫0tAyn−1s+fs,yn−1s,yn−1s−τ2ds+16σ¯2E^∫0tgs,yn−1s,yn−1s−τ2ds+4σ¯4tE^∫0ths,yn−1s,yn−1s−τ2ds.
Using inequality (21) and the Cauchy inequality, we obtain(42)E^ynt2≤4E^y02+8tE^∫0tAyn−1s2+fs,yn−1s,yn−1s−τ2ds+16σ¯2E^∫0tgs,yn−1s,yn−1s−τ2ds+4σ¯4tE^∫0ths,yn−1s,yn−1s−τ2ds≤4Ey02+8TE^∫0tG1+A2+G1yn−1s2+G1yn−1s−τ2ds+16σ¯2E^∫0tG1+G1yn−1s2+G1yn−1s−τ2ds+4σ¯4TE^∫0tG1+G1yn−1s2+G1yn−1s−τ2ds≤C1+8T+16σ¯2+4σ¯4TE^∫0tA2+G1yn−1s2+G1yn−1s−τ2ds≤C1+8T+16σ¯2+4σ¯4T∫0tA2+2G1E^sup0≤r≤syn−1r2+G1E^y02ds≤C2+4A2+2G12T+4σ¯2+σ¯4T∫0tE^sup0≤r≤syn−1r2ds≤C2+4A2+2G12T+4σ¯2+σ¯4T∫0tE^yn−1s2ds.where C1=4E^y02+8T+16σ¯2+4σ¯4TTG1 and C2=C1+8T+16σ¯2+4σ¯4TG1TE^y02.
Hence, for anyk≥1, we derive that(43)max1≤n≤kE^ynt2≤C2+4A2+2G12T+4σ¯2+σ¯4T∫0tmax1≤n≤kE^yn−1s2ds.
Note that(44)max1≤n≤kE^yn−1s2≤maxE^y02,max1≤n≤kE^yns2≤E^y02+max1≤n≤kE^yns2.
We have(45)max1≤n≤kE^ynt2≤C2+4A2+2G12T+4σ¯2+σ¯4T∫0tE^y02+max1≤n≤kE^yns2ds≤C3+4A2+2G12T+4σ¯2+σ¯4T∫0tmax1≤n≤kE^yns2ds,where C3=C2+4TA2+2G12T+4σ¯2+σ¯4TE^y02. It follows from the Gronwall inequality that(46)max1≤n≤kE^ynt2≤C3e4A2+2G12T+4σ¯2+σ¯4T.
Sincek is arbitrary, we can have(47)E^ynt2≤C3e4A2+2G12T+4σ¯2+σ¯4T,for all0≤t≤T,n≥1.
This shows thatynt∈MG20,T;Rd.
Using the elementary inequality, we obtain(48)y1t−y0t2=y1t−y02≤3∫0tAy0s+fs,y0,y0s−τds2+3∫0tgs,y0s,y0s−τdωs2+3∫0ths,y0s,y0s−τdωs2.
Taking the G-expectation on (48), it follows from Propositions 2 and 3 and the properties of G-Ito^ integral that(49)E^y1t−y0t2≤3E^∫0tAy0s+fs,y0s,y0s−τds2+3E^∫0tgs,y0s,y0s−τdωs2+3E^∫0ths,y0s,y0s−τdωs2≤3tE^∫0tAy0s+fs,y0s,y0s−τ2ds+12σ¯2∫0tgs,y0s,y0s−τ2ds+3σ¯4t∫0ths,y0s,y0s−τ2ds≤6t+12σ¯2+3σ¯4tE^∫0tG1+2G1+A2y0s2ds≤6t+12σ¯2+3σ¯4tG1t+6t+12σ¯2+3σ¯4t2G1+A2tE^y02,that is,(50)max0≤s≤tE^y1s−y0s2≤6t+12σ¯2+3σ¯4tG1t+6t+12σ¯2+3σ¯4t2G1+A2tE^y02.
Takingt=T,(51)max0≤s≤tE^y1s−y0s2≤6t+12σ¯2+3σ¯4tG1T+6t+12σ¯2+3σ¯4t2G1+A2TE^y02≔C¯.
By the same ways as earlier, we compute(52)E^y2t−y1t2≤6tE^∫0tAy1s−Ay0s2+fs,y1,y1s−τ−fs,y0s,y0s−τ2ds+12σ¯2∫0tgs,y1s,y1s−τ−gs,y0s,y0s−τ2ds+3σ¯4t∫0ths,y1s,y1s−τ−hs,y0s,y0s−τ2ds,and thus, we derive that(53)E^sup0≤s≤ty2s−y1s2≤6tE^∫0tA2|y1s−y0s|2+L1y1s−y0s2+y1s−τ−y0s−τ2ds+12σ¯2∫0tL1|y1s−y0s|2+y1s−τ−y0s−τ2ds+3σ¯4t∫0tL1y1s−y0s2+y1s−τ−y0s−τ2ds≤32T+4σ¯2+σ¯4TE^∫0tA2+2L1sup0≤r≤sy1r−y0r2ds≤32T+4σ¯2+σ¯4TA2+2L1tC¯.
Similarly,(54)E^sup0≤s≤ty3s−y2s2≤32T+4σ¯2+σ¯4TE^∫0tA2+2L1sup0≤r≤sy2r−y1r2ds≤32T+4σ¯2+σ¯4TA2+2L1E^∫0t32T+4σ¯2+σ¯4TA2+2L1sC¯ds≤322T+4σ¯2+σ¯4T2A2+2L12t2C¯2!,continuing this process to find that(55)E^sup0≤s≤ty4s−y3s2≤32T+4σ¯2+σ¯4TE^∫0tA2+2L1sup0≤s≤ty3r−y2r2ds≤32T+4σ¯2+σ¯4TA2+2L1E^∫0t322T+4σ¯2+σ¯4T2A2+2L12s2C¯ds≤332T+4σ¯2+σ¯4T3A2+2L13t3C¯3!.
Now, we claim that, for alln≥0,(56)E^sup0≤s≤tyn+1s−yns2≤C¯Mtnn!,0≤t≤T,where M=32T+4σ¯2+σ¯4TA2+2L1.
Whenn=0,1,2,3, inequality (56) holds. We suppose that (56) holds for some n, now to check (56) for n+1. In fact,(57)E^sup0≤s≤tyn+2s−yn+1s2≤32T+4σ¯2+σ¯4TE^∫0tA2+2L1sup0≤r≤syn+1r−ynr2ds=M∫0tE^A2+2L1sup0≤r≤syn+1r−ynr2ds.
By induction and (56),(58)E^sup0≤s≤tyn+2s−yn+1s2≤M∫0tC¯Msnn!ds=C¯Mtn+1n+1!.
It is easy to see that (56) holds for n+1. Therefore, by induction, (56) holds for all n≥0.
Next, to verify thatynt,n≥0 converge to yt at the sense of LG2 and probability 1 on MG20,T;Rd, moreover, yt is the solution of (7). For (56), taking t=T,(59)E^sup0≤t≤Tyn+1t−ynt2≤C¯MTnn!.
By Chebyshev’s inequality,(60)E^sup0≤t≤Tyn+1t−ynt>12n≤C¯2MTnn!.
Since∑n=0∞C¯2MTn/n!<∞, the Borel-Cantelli lemma yields that there exists a positive integer n0 such that(61)sup0≤t≤Tyn+1t−ynt≤12nwhenevern≥n0.
It follows that, with probability 1, the partial sums,(62)y0t+∑i=0n−1yi+1t−yit=ynt,are convergent uniformly in t∈0,T. Denote the limit by yt. Clearly, yt is continuous and ℱt-adapted. On the other hand, one sees from (56) that, for every t, yntn≥1 is a Cauchy sequence in LG2 as well. Hence, we also have ynt⟶yt in LG2; that is,(63)E^ynt−yt2⟶0,n⟶∞.
Lettingn⟶∞ in (47) then yields that(64)E^yt2≤C3e4A2+2G12T+4σ¯2+σ¯4T,for all0≤t≤T.
Therefore,y⋅∈M20,T;Rd. It remains to show that yt satisfies the G-SLSDDE in integral form. Note that(65)E^∫0tAyns+fs,yns,yns−τds−∫0tAys+fs,ys,ys−τds2+E^∫0tgs,yns,yns−τdωs−∫0tgs,ys,ys−τdωs2+E^∫0ths,yns,yns−τdωs−∫0ths,ys,ys−τdωs2≤tE^∫0tAyns+fs,yns,yns−τ−Ays−fs,ys,ys−τ2ds+4σ¯2E^∫0tgs,yns,yns−τ−gs,ys,ys−τ2ds+σ¯4tE^∫0ths,yns,yns−τ−hs,ys,ys−τ2ds≤L11+σ¯4T+4σ¯2A2L1+2∫0tE^sup0≤r≤synr−yr2ds≤L11+σ¯4T+4σ¯2A2L1+2∫0tE^yns−ys2ds.
Noting that sequenceynt,n≥0 uniformly converges on 0,T, it means that, for any given ε>0, there exists a positive integer n0 such that as n≥n0, for any 0,T, one then deduces that E^ynt−yt2<ε. Furthermore, we obtain(66)E^∫0tAyns+fs,yns,yns−τds−∫0tAys+fs,ys,ys−τds2+E^∫0tgs,yns,yns−τdωs−∫0tgs,ys,ys−τdωs2+E^∫0ths,yns,yns−τdωs−∫0ths,ys,ys−τdωs2≤L11+σ¯4T+4σ¯2A2L1+2∫0tE^yns−ys2ds≤L11+σ¯4T+4σ¯2A2L1+2Tε.
In other words, fort∈0,T, we have(67)∫0tAyns+fs,yns,yns−τds⟶∫0tAys+fs,ys,ys−τdsinLG2,∫0tgs,yns,yns−τds⟶∫0tgs,ys,ys−τdωsinLG2,∫0ths,yns,yns−τds⟶∫0ths,ys,ys−τdωsinLG2.
For0≤t≤T, taking limits on both sides of (47), we obtain(68)limn⟶∞ynt=y0+limn⟶∞∫0tAyns+fs,yns,yns−τds+limn⟶∞∫0tgs,yns,yns−τdωs+limn⟶∞∫0ths,yns,yns−τdωs,that is,(69)yt=y0+∫0tAys+fs,ys,ys−τds+∫0tgs,ys,ys−τdωs+∫0ths,ys,ys−τdωs,0≤t≤T.
The aformentioned expression demonstrates thatyt is the solution of (7). So far, the existence of Theorem 1 is completed.


## 4. Exponential Euler Method and the Numerical Analysis

We can now give the numerical solution for the G-SLSDDE (7). In order to avoid the storage problem and improve the convergence order, we introduce the exponential Euler scheme.To formulate the grid, lett0 be an arbitrary but fixed positive number, define tn=t0+nh, n=0,1,2,…, h>0 is the step size which satisfies h=τ/m, m is a positive integer, and the solution of (19) at tn+1=tn+h has the form(70)ytn+1=eAhnytn+∫tntn+1eAsftn,ys,ys−τds+∫tntn+1eAsgtn,ys,ys−τdωs+∫tntn+1eAshtn,ys,ys−τdωs.Approximating the functionsf, g, and h with the integral by left rectangle formula at the known values ftn,ytn,ytn−τ, gtn,ytn,ytn−τ, and htn,ytn,ytn−τ only leads to the exponential Euler method:(71)yn+1=eAhnyn+eAhnftn,yn,yn−mhn+eAhngtn,yn,yn−mΔωn+eAhnhtn,yn,yn−mΔωn.where yn is an approximation to ytn and yn−m is an approximation to ytn−τ. The increments Δωn=ωtn−ωtn−1 and Δωn=ωtn−ωtn−1. We assume that yn is ℱtn−measurable at the mesh point tn.Letu¯=⌊u/h⌋h with ⌊y⌋ denoting the largest integer which is smaller than y, and zt=∑k=0∞ykItk,tk+1t with Itk,tk+1t denoting the indicator function on tk,tk+1; namely, Itk,tk+1=1 if t∈tk,tk+1 or 0 otherwise. Then, we can extend the discrete exponential Euler method (71) to the continuous one as follows:(72)yt^=eAty0+∫0teAt−u¯fu¯,zu,u−τdu+∫0teAt−u¯gu¯,zu,zu−τdωu+∫0teAt−u¯hu¯,zu,zu−τdωu.It is not difficult to see thatytn^=yn for n=0,1,2,⋯, that is, the continuous extension yt^ coincides with the discrete numerical solutions at the mesh points.


### 4.1. Convergence of the Exponential Euler Method

Now, we show the convergence of the exponential Euler approximate solution to the exact solution for system (7). To obtain this result, we first need to show the following several definitions.Definition 6.
(1)
The global error for exponential Euler method is defined as follows:(73)εn=ytn−yn.(2)
For fixedT<∞, the approximation yn is convergent in the mean-square sense on mesh points, with strong order p if(74)max1≤n≤NE^εn21/2≤Chp,ash⟶0,where C is a positive constant.
We will use the following lemma to analyze the convergence of the exponential Euler method.Lemma 3.
Under the Lipschitz condition (20), there exists a constant C˜1>0, such that the analytic solution yt of Equation (7) and the numerical solution yt^ of continuous exponential Euler approximate (72) satisfy the following inequality:(75)E^sup0≤t≤Tyt2∨E^sup0≤t≤Tyt^2≤C˜1,where C˜1 is independent of h, and(76)C˜1=max4e2ATE^y02+T+4σ¯2+σ¯TTG1e8e2ATT+4σ¯2+σ¯TTG1,C3e4A2+2G12T+4σ¯2+σ¯T.Proof.
The detailed proofs are similar to the proofs in [17] which are omitted here.Theorem 2.
Under the Lipschitz condition (20), there exists a nonnegative constant G2, such that, for any s,t∈0,T and t>s,(77)ft,x,y−fs,x,y2∨gt,x,y−gs,x,y2∨ht,x,y−gs,x,y2≤G21+x2+y2s−t.
The numerical solution produced by the continuous exponential Euler method (72) converges to the analytical solution of equation (7) in MS sense with the strong order 1/2; that is, there exists a positive constant C, such that(78)Esup0≤t≤Tyt−yt^2≤Ch,ash⟶0.Proof.
Taking the difference between (19) and (72) and squaring its both sides, we can get(79)yt−yt^2=∫0teAt−ufu,yu,yu−τdu−∫0teAt−u¯fu¯,zu,zu−τdu+∫0teAt−ugu,yu,yu−τdωu−∫0teAt−u¯gu¯,zu,zu−τdωu−τ2+∫0teAt−uhu,yu,yu−τdωu−∫0teAt−u¯hu¯,zu,zu−τdωu−τ|2.
With the help of the elementary inequality and Propositions2 and 3, we can get(80)yt−yt^2≤3∫0teAt−ufu,yu,yu−τdu−∫0teAt−u¯fu¯,zu,zu−τdu2+3∫0teAt−ugu,yu,yu−τdωu−∫0teAt−u¯gu¯,zu,zu−τdωu2+3∫0teAt−uhu,yu,yu−τdωu−∫0teAt−u¯hu¯,zu,zu−τdωu2≤9T∫0t|eAt−ufu,yu,yu−τ−eAt−u¯fu,yu,yu−τ|2du+9T∫0teAt−u¯fu,yu,yu−τ−eAt−u¯fu¯,yu,yu−τ2du+9T∫0teAt−u¯fu¯,yu,yu−τ−eAt−u¯fu¯,zu,zu−τ2du+36σ¯2∫0teAt−ugu,yu,yu−τ−eAt−u¯gu,yu,yu−τ2du+36σ¯2∫0teAt−u¯gu,yu,yu−τ−eAt−u¯gu¯,yu,yu−τ2du+36σ¯2∫0teAt−u¯gu¯,yu,yu−τ−eAt−u¯gu¯,zu,zu−τ2du+9σ¯4T∫0teAt−uhu,yu,yu−τ−eAt−u¯hu,yu,yu−τ2du+9σ¯4T∫0teAt−u¯hu,yu,yu−τ−eAt−u¯hu¯,yu,yu−τ2du+9σ¯4T∫0teAt−u¯hu¯,yu,yu−τ−eAt−u¯hu¯,zu,zu−τ2du.
Taking G-expectation on both sides of (80), for arbitrary 0≤t1≤T, we arrive at(81)E^sup0≤t≤t1yt−yt^2≤9TE^sup0≤t≤t1∫0teAt−ufu,yu,yu−τ−eAt−u¯fu,yu,yu−τ2du+9TE^sup0≤t≤t1∫0teAt−u¯fu,yu,yu−τ−eAt−u¯fu¯,yu,yu−τ2du+9TE^sup0≤t≤t1∫0teAt−u¯fu¯,yu,yu−τ−eAt−u¯fu¯,zu,zu−τ2du+36σ¯2E^sup0≤t≤t1∫0teAt−ugu,yu,yu−τ−eAt−u¯gu,yu,yu−τ2du+36σ¯2E^sup0≤t≤t1∫0teAt−u¯gu,yu,yu−τ−eAt−u¯gu¯,yu,yu−τ2du+36σ¯2E^sup0≤t≤t1∫0teAt−u¯gu¯,yu,yu−τ−eAt−u¯gu¯,zu,zu−τ2du+9σ¯4TE^sup0≤t≤t1∫0teAt−uhu,yu,yu−τ−eAt−u¯hu,yu,yu−τ2du+9σ¯4TE^sup0≤t≤t1∫0teAt−u¯hu,yu,yu−τ−eAt−u¯hu¯,yu,yu−τ2du+9σ¯4TE^sup0≤t≤t1∫0teAt−u¯hu¯,yu,yu−τ−eAt−u¯hu¯,zu,zu−τ2du≤9TJ1t+9TJ2t+9TJ3t+36σ¯2J4t+36σ¯2J5t+36σ¯2J6t+9σ¯4TJ7t+9σ¯4TJ8t+9σ¯4TJ9t,where(82)J1t=E^sup0≤t≤t1∫0teAt−ufu,yu,yu−τ−eAt−u¯fu,yu,yu−τ2du,J2t=E^sup0≤t≤t1∫0teAt−u¯fu,yu,yu−τ−eAt−u¯fu¯,yu,yu−τ2du,J3t=E^sup0≤t≤t1∫0teAt−u¯fu¯,yu,yu−τ−eAt−u¯fu¯,zu,zu−τ2du,J4t=E^sup0≤t≤t1∫0teAt−ugu,yu,yu−τ−eAt−u¯gu,yu,yu−τ2du,J5t=E^sup0≤t≤t1∫0teAt−u¯gu,yu,yu−τ−eAt−u¯gu¯,yu,yu−τ2du,J6t=E^sup0≤t≤t1∫0teAt−u¯gu¯,yu,yu−τ−eAt−u¯gu¯,zu,zu−τ2du,J7t=E^sup0≤t≤t1∫0teAt−uhu,yu,yu−τ−eAt−u¯hu,yu,yu−τ2du,J8t=E^sup0≤t≤t1∫0teAt−u¯hu,yu,yu−τ−eAt−u¯hu¯,yu,yu−τ2du,J9t=E^sup0≤t≤t1∫0teAt−u¯hu¯,yu,yu−τ−eAt−u¯hu¯,zu,zu−τ2du.
To estimate (81), we need to estimate Jiti=1,2,3,4,5,7,8,9.
ForJ1t, we have that(83)J1t≤E^sup0≤t≤t1∫0teAt−u−eAt−u¯2fu,yu,yu−τ2du=E^sup0≤t≤t1∫0teAt−u2I−eAu−u¯2fu,yu,yu−τ2du.
Due to the linear growth condition (21) and eAh−I≤AheAT, inequality (83) gives(84)E^sup0≤t≤t1∫0teAt−u−eAt−u¯2fu,yu,yu−τ2du≤G1E^sup0≤t≤t1∫0te2At−uI−eAu−u¯21+yu2+yu−τ2du≤G1e2ATTA2h2e2AT1+2C˜1.
Using (78) and Lipschitz condition (20), it follows from J2t and J3t that(85)J2t≤G2e2ATE^sup0≤t≤t1∫0t1+yu2+yu−τ2u−u¯du≤G2e2ATTh1+2C˜1,(86)J3t≤L1e2|A|TE^sup0≤t≤t1∫0tyu−zu2+yu−τ−zu−τ2du≤2L1e2|A|TT∫0t1E^sup0≤u≤tyu−yu^2dt.
Similar to estimatesJ1t, J2t, and J3t, one obtains(87)J4t≤G1e2ATTA2h2e2AT1+2C˜1,(88)J5t≤G2e2ATTh1+2C˜1,(89)J6t≤2L1e2ATT∫0t1E^sup0≤u≤tyu−yu^|2dt,(90)J7t≤G1e2ATTA2h2e2AT1+2C˜1,(91)J8t≤G2e2ATTh1+2C˜1,(92)J9t≤2L1e2ATT∫0t1E^sup0≤u≤tyu−yu^2dt.
Substituting (84)–(92) into (81) and rearranging (81), we obtain(93)E^sup0≤t≤t1yt−yt^2≤9T+36σ¯2+9Tσ¯4G1e2ATTA2h2e2AT1+2C˜1,(94)+9T+36σ¯2+9Tσ¯4G2e2ATTh1+2C˜1+29T+36σ¯2+9Tσ¯4L1e2ATT∫0t1E^sup0≤u≤tyu−yu^|2dt≤9T+36σ¯2+9Tσ¯4e2ATThG1A2he2AT+G21+2C˜1+29T+36σ¯2+9Tσ¯4L1e2ATT∫0t1E^sup0≤u≤tyu−yu^2dt.
By Gronwall’s inequality, it gives(95)E^sup0≤t≤Tyt−yt^2≤9T+36σ¯2+9Tσ¯4e2AThT+4G1A2he2AT+G21+2C˜1⋅e29T+36σ¯2+9Tσ¯4L1e2|A|TT,and then, letting h⟶0, one can draw the conclusion. The proof is completed.
Theorem2 shows that if the coefficients of f, g, and h obey the Lipschitz condition, in addition to the conditions imposed in Lemma 3, then the exponential Euler approximate solution converges to the exact solution for system (7).


### 4.2. The Mean-Square Stability

In this section, we are in a position to explore the exponential stability of the exact solution and the numerical approximation. For this purpose, we further assume thatft,0,0=0, gt,0,0=0, and ht,0,0=0. Therefore, system (7) admits a trivial solution. Moreover, from condition (20), it is easy to get(96)ft,x,y2∨gt,x,y2∨ht,x,y2≤L1x2+y2,∀x,y∈Rd.In the following, we first give some necessary assumptions and definitions for the mean-square exponential stability.Definition 7.
Equation (7) is said to be mean-square exponentially stable if there exist positive constants v and C, which is dependent on the initial data y0t∈Cℱ0b0,T;Rd and independent of t, such that(97)E^yt2≤Ce−vt,t≥0.
For convenience, we introduce the logarithmic normμA proposed in [38].Definition 8.
The logarithmic normμA of A is defined by(98)μA=limΔ⟶0+I+ΔA−1Δ.
In particular, if⋅ is an inner product norm, μA can also be written as(99)μA=maxy≠0Ay,yy2.Theorem 3.
Assume thatf, g, and h satisfy the Lipschitz condition (20). If there exist positive constants λ1 and λ2, such that(100)2xTft,x,y≤−λ1x2+λ2y2,x,y∈Rd,holds, then the analytical solution of equation (7) is mean-square exponentially stable under the condition(101)λ1−2μA−λ2−6σ¯4L1−4σ¯4L1>0.Proof.
LetVt,yt=1+yt2=1+yTtyt. By the matrix derivative rule and the Ito^ formula, we can derive that(102)dVt,yt=Vtt,yt+Vyt,ytft,yt,yt−τdt+Vyt,ytgt,yt,yt−τdωt+Vyt,ytht,yt,yt−τ+12tracegTVyyt,yt,yt−τgdωt.
The aforementioned equation is equivalent to the following stochastic integral equation:(103)Vt,yt=V0,y0+∫0tVss,ys+Vys,ysfs,ys,ys−τds+∫0tVys,ysgs,ys,ys−τdωs+∫0tVys,yshs,ys,ys−τ+12tracegTVyys,ys,ys−τgdωs,that is,(104)1+yt2=1+y02+2∫0tys,Ays+fs,ys,ys−τds+2∫0tys,gs,ys,ys−τdωs+2∫0tys,hs,ys,ys−τdωs+∫0tgs,ys,ys−τ2dωs.
Taking G-expectation on both sides of (101), we get(105)E^1+yt2=1+y02+2E^∫0tys,Ays+fs,ys,ys−τds+2E^∫0tys,gs,ys,ys−τdωs+2E^∫0tys,hs,ys,ys−τdωs+E^∫0tgs,ys,ys−τ2dωs.
With the condition in (100), the inequality in (105) obtains(106)E^sup0≤s≤t1+ys2≤1+y02+E^∫0t2μA−λ1ys2+λ2ys−τ2ds+2E^sup0≤s≤t∫0syu,gu,yu,yu−τdωu+2E^sup0≤s≤t∫0syu,hu,yu,yu−τdωu+E^sup0≤s≤t∫0sgu,yu,yu−τ2dωu.
Combining Proposition3 with Lemma 1, we have(107)2E^sup0≤s≤t∫0syu,gu,yu,yu−τdωu≤2σ¯2E^∫0tys2gs,ys,ys−τ2ds1/2≤2σ¯2E^sup0≤s≤tys2∫0tgs,ys,ys−τ2ds1/2≤12E^sup0≤s≤t1+ys2+2σ¯4L1E^∫0tys2+ys−τ2ds.
Similarly, we have(108)2E^sup0≤s≤t∫0syu,hu,yu,yu−τdωu≤2σ¯4E^∫0styshs,ys,ys−τds≤σ¯4E^∫0t2L14yshs,ys,ys−τL14ds≤σ¯4E^∫0tL1ys2+hs,ys,ys−τ2L1ds≤σ¯4L1E^∫0tys2+ys2+ys−τ2ds≤2σ¯4L1E^∫0tys2+ys−τ2ds.and(109)E^sup0≤s≤t∫0sgu,yu,yu−τ2dωu≤σ¯4L1E^∫0tys2+ys−τ2ds.
Substituting (107)–(109) into (106), we obtain(110)E^sup0≤s≤t1+ys2≤1+y02+E^∫0t2μA−λ1yu2+λ2yu−τ2du+12E^sup0≤s≤t1+ys2+2σ¯4L1E^∫0tys2+ys−τ2ds+2σ¯4L1E^∫0tys2+ys−τ2ds+σ¯4L1E^∫0tys2+ys−τ2ds,that is,(111)E^sup0≤s≤t1+ys2≤21+y02+22μA−λ1+λ2+6σ¯4L1+4σ¯4L1E^∫0tsup0≤s≤tys2ds.
From the Gronwall inequality, we have(112)E^sup0≤s≤tys2≤21+y02e22μA−λ1+λ2+6σ¯4L1+4σ¯4L1t≤Ce−vt.where C=21+y02 and v=2λ1−2μA−λ2−6σ¯4L1−4σ¯4L1. The proof is completed.Then the question is, will the exponential Euler method reproduce the stability of analytical solutions of equation (7) under the Lipschitz condition? In order to answer this question, we introduce the following definition of mean-square stability from [39] at first and present the stability theorem of the exponential Euler approximation.Definition 9.
A numerical method is said to be asymptotically mean-square stable (with respect to a given G-SLSDDE) if(113)limn⟶∞E^yn2=0.Theorem 4.
Let (H1) hold. Assume that there exist positive constantsλ3 and λ4, such that(114)2σ¯2xTht,x,y≤−λ3x2+λ4y2,x,y∈Rd.
If the inequalitiesL1σ¯2−λ1−λ3<0 and 2μA+λ2+λ4+λ1+λ3<0 hold, then the exponential Euler method is asymptotically mean-square stable for all(115)hn<h0=1/2λ1+1/2λ3−1/2L1σ¯2L1+L1σ¯4+L1σ¯2.Proof.
Squaring both sides of (71), we have(116)yn+12=e2μAhnyn2+f2tn,yn,yn−mhn2+g2tn,yn,yn−mΔωn2+h2tn,yn,yn−mΔωn2+2e2μAhnynTftn,yn,yn−mhn+2e2μAhnynTgtn,yn,yn−mΔωn+2e2μAhnynThtn,yn,yn−mΔωn+2e2μAhnftn,yn,yn−mTgtn,yn,yn−mhnΔωn+2e2μAhnftn,yn,yn−mThtn,yn,yn−mhnΔωn+2e2μAhngtn,yn,yn−mThtn,yn,yn−mΔωnΔωn.
Taking G-expectation on both sides of (116) and considering that yk and yk−m are ℱtk-measurable, by Proposition 3, one can get(117)E^g2tk,yk,yk−mΔωk2−σ¯2hk=E^E^g2tk,yk,yk−mΔωk2−σ¯2hk|ℱtk=E^g2tk,yk,yk−mE^Δωk2−σ¯2hk|ℱtk=0,E^ykTgtk,yk,yk−mΔωk=E^E^ykTgtk,yk,yk−mΔωk|ℱtk=E^ykTgtk,yk,yk−mE^Δωk|ℱtk=0.
By Proposition2, one obtains(118)E^h2tk,yk,yk−mΔωk2−σ¯4hk2=E^E^h2tk,yk,yk−mΔωk2−σ¯4hk2|ℱtk=E^g2tk,yk,yk−mE^Δωk2−σ¯4hk2|ℱtk≤0,E^ykThtk,yk,yk−mΔωk=E^E^ykTgtk,yk,yk−mΔωk|ℱtk=E^ykThtk,yk,yk−mE^Δωk|ℱtk≤σ¯2hkE^ykThtk,yk,yk−m,E^gTtk,yk,yk−mhtk,yk,yk−mΔωkΔωk=E^E^gTtk,yk,yk−mhtk,yk,yk−mΔωkΔωk|ℱtk=E^gTtk,yk,yk−mhtk,yk,yk−mE^ΔωkΔωk|ℱtk≤σ¯2hkE^gTtk,yk,yk−mhtk,yk,yk−mE^Δωk|ℱtk=0,E^fTtk,yk,yk−mhtk,yk,yk−mhkΔωk=E^E^fTtk,yk,yk−mhtk,yk,yk−mhkΔωk|ℱtk=hkE^fTtk,yk,yk−mhtk,yk,yk−mE^Δωk|ℱtk≤σ¯2hk2E^fTtk,yk,yk−mhtk,yk,yk−m.
Then, it follows from (100), (114), and (99) that(119)E^yn+12=e2μAhn|E^yn2+E^f2tn,yn,yn−mhn2+E^g2tn,yn,yn−mσ¯2hn+E^h2tn,yn,yn−mσ¯4hn2+2ynTftn,yn,yn−mhn+2σ¯2hnE^ynThtn,yn,yn−m+2σ¯2hn2E^ftn,yn,yn−mThtn,yn,yn−m|≤e2μAhn|E^yn2+E^L1yn2+L1yn−m2hn2+E^L1yn2+L1yn−m2σ¯2hn+E^L1yn2+L1yn−m2σ¯4hn2+−λ1E^yn2+λ2E^yn−m2hn+hnE^−λ3yn2+λ4yn−m2+2σ¯2hn2E^L1yn2+L1yn−m2|=e2μAhn|1+L1hn2+L1σ¯2hn+L1σ¯4hn2−λ1hn−λ3hn+L1σ¯2hn2E^yn2+L1hn2+L1σ¯2hn+L1σ¯4hn2+λ2hn+λ4hn+L1σ¯2hn2E^yn−m2=A1E^yn2+A2E^yn−m2,where A1=e2μAhn1+L1σ¯2−λ1−λ3hn+L1+L1σ¯4+L1σ¯2hn2 and A2=e2μAhnL1σ¯2+λ2+λ4hn+L1+L1σ¯4+L1σ¯2hn2.
ConsideringL1σ¯2−λ1−λ3<0, for any hn<h0=1/2λ1+1/2λ3−1/2L1σ¯2/L1+L1σ¯4+L1σ¯2, the following inequalities hold:(120)A1≤e2μAhn1+12L1σ¯2−12λ1−12λ3hn,(121)A1+A2≤A1+A2=e2μAhn1+12L1σ¯2−12λ1−12λ3hn+e2μAhnL1σ¯2+λ2+λ4hn+L1+L1σ¯4+L1σ¯2hn2≤e2μAhn1+12λ1+12λ3−12L1σ¯2hn+e2μAhnL1σ¯2+λ2+λ4+12λ1+12λ3−12L1σ¯2hn≤e2μAhn1+λ2+λ4+λ1+λ3hn.
Therefore,(122)E^yn+12≤e2μAhn1+λ2+λ4+λ1+λ3hnmaxE^yn2,E^yn−m2≤e2μA+λ2+λ4+λ1+λ3hnmaxE^yn2,E^yn−m2.
The condition2μA+λ2+λ4+λ1+λ3<0 gives that e2μA+λ2+λ4+λ1+λ3hn<1; it is not difficult to see that E^yn2≤E^yn−m2; therefore,(123)E^yn+12≤e2μA+λ2+λ4+λ1+λ3hnE^yn−m2≤e22μA+λ2+λ4+λ1+λ3hnE^yn−2m2⋮≤en−2/m2μA+λ2+λ4+λ1+λ3hnE^y02,and hence(124)limn⟶∞E^yn2=0.
The above theorem gives the sufficient conditions for keeping mean-square stability by exponential Euler method for equation (7). On the premise of its stability, it is found that the stability and step size of exponential Euler method depend on the norm of A.


## 4.1. Convergence of the Exponential Euler Method

Now, we show the convergence of the exponential Euler approximate solution to the exact solution for system (7). To obtain this result, we first need to show the following several definitions.Definition 6.
(1)
The global error for exponential Euler method is defined as follows:(73)εn=ytn−yn.(2)
For fixedT<∞, the approximation yn is convergent in the mean-square sense on mesh points, with strong order p if(74)max1≤n≤NE^εn21/2≤Chp,ash⟶0,where C is a positive constant.
We will use the following lemma to analyze the convergence of the exponential Euler method.Lemma 3.
Under the Lipschitz condition (20), there exists a constant C˜1>0, such that the analytic solution yt of Equation (7) and the numerical solution yt^ of continuous exponential Euler approximate (72) satisfy the following inequality:(75)E^sup0≤t≤Tyt2∨E^sup0≤t≤Tyt^2≤C˜1,where C˜1 is independent of h, and(76)C˜1=max4e2ATE^y02+T+4σ¯2+σ¯TTG1e8e2ATT+4σ¯2+σ¯TTG1,C3e4A2+2G12T+4σ¯2+σ¯T.Proof.
The detailed proofs are similar to the proofs in [17] which are omitted here.Theorem 2.
Under the Lipschitz condition (20), there exists a nonnegative constant G2, such that, for any s,t∈0,T and t>s,(77)ft,x,y−fs,x,y2∨gt,x,y−gs,x,y2∨ht,x,y−gs,x,y2≤G21+x2+y2s−t.
The numerical solution produced by the continuous exponential Euler method (72) converges to the analytical solution of equation (7) in MS sense with the strong order 1/2; that is, there exists a positive constant C, such that(78)Esup0≤t≤Tyt−yt^2≤Ch,ash⟶0.Proof.
Taking the difference between (19) and (72) and squaring its both sides, we can get(79)yt−yt^2=∫0teAt−ufu,yu,yu−τdu−∫0teAt−u¯fu¯,zu,zu−τdu+∫0teAt−ugu,yu,yu−τdωu−∫0teAt−u¯gu¯,zu,zu−τdωu−τ2+∫0teAt−uhu,yu,yu−τdωu−∫0teAt−u¯hu¯,zu,zu−τdωu−τ|2.
With the help of the elementary inequality and Propositions2 and 3, we can get(80)yt−yt^2≤3∫0teAt−ufu,yu,yu−τdu−∫0teAt−u¯fu¯,zu,zu−τdu2+3∫0teAt−ugu,yu,yu−τdωu−∫0teAt−u¯gu¯,zu,zu−τdωu2+3∫0teAt−uhu,yu,yu−τdωu−∫0teAt−u¯hu¯,zu,zu−τdωu2≤9T∫0t|eAt−ufu,yu,yu−τ−eAt−u¯fu,yu,yu−τ|2du+9T∫0teAt−u¯fu,yu,yu−τ−eAt−u¯fu¯,yu,yu−τ2du+9T∫0teAt−u¯fu¯,yu,yu−τ−eAt−u¯fu¯,zu,zu−τ2du+36σ¯2∫0teAt−ugu,yu,yu−τ−eAt−u¯gu,yu,yu−τ2du+36σ¯2∫0teAt−u¯gu,yu,yu−τ−eAt−u¯gu¯,yu,yu−τ2du+36σ¯2∫0teAt−u¯gu¯,yu,yu−τ−eAt−u¯gu¯,zu,zu−τ2du+9σ¯4T∫0teAt−uhu,yu,yu−τ−eAt−u¯hu,yu,yu−τ2du+9σ¯4T∫0teAt−u¯hu,yu,yu−τ−eAt−u¯hu¯,yu,yu−τ2du+9σ¯4T∫0teAt−u¯hu¯,yu,yu−τ−eAt−u¯hu¯,zu,zu−τ2du.
Taking G-expectation on both sides of (80), for arbitrary 0≤t1≤T, we arrive at(81)E^sup0≤t≤t1yt−yt^2≤9TE^sup0≤t≤t1∫0teAt−ufu,yu,yu−τ−eAt−u¯fu,yu,yu−τ2du+9TE^sup0≤t≤t1∫0teAt−u¯fu,yu,yu−τ−eAt−u¯fu¯,yu,yu−τ2du+9TE^sup0≤t≤t1∫0teAt−u¯fu¯,yu,yu−τ−eAt−u¯fu¯,zu,zu−τ2du+36σ¯2E^sup0≤t≤t1∫0teAt−ugu,yu,yu−τ−eAt−u¯gu,yu,yu−τ2du+36σ¯2E^sup0≤t≤t1∫0teAt−u¯gu,yu,yu−τ−eAt−u¯gu¯,yu,yu−τ2du+36σ¯2E^sup0≤t≤t1∫0teAt−u¯gu¯,yu,yu−τ−eAt−u¯gu¯,zu,zu−τ2du+9σ¯4TE^sup0≤t≤t1∫0teAt−uhu,yu,yu−τ−eAt−u¯hu,yu,yu−τ2du+9σ¯4TE^sup0≤t≤t1∫0teAt−u¯hu,yu,yu−τ−eAt−u¯hu¯,yu,yu−τ2du+9σ¯4TE^sup0≤t≤t1∫0teAt−u¯hu¯,yu,yu−τ−eAt−u¯hu¯,zu,zu−τ2du≤9TJ1t+9TJ2t+9TJ3t+36σ¯2J4t+36σ¯2J5t+36σ¯2J6t+9σ¯4TJ7t+9σ¯4TJ8t+9σ¯4TJ9t,where(82)J1t=E^sup0≤t≤t1∫0teAt−ufu,yu,yu−τ−eAt−u¯fu,yu,yu−τ2du,J2t=E^sup0≤t≤t1∫0teAt−u¯fu,yu,yu−τ−eAt−u¯fu¯,yu,yu−τ2du,J3t=E^sup0≤t≤t1∫0teAt−u¯fu¯,yu,yu−τ−eAt−u¯fu¯,zu,zu−τ2du,J4t=E^sup0≤t≤t1∫0teAt−ugu,yu,yu−τ−eAt−u¯gu,yu,yu−τ2du,J5t=E^sup0≤t≤t1∫0teAt−u¯gu,yu,yu−τ−eAt−u¯gu¯,yu,yu−τ2du,J6t=E^sup0≤t≤t1∫0teAt−u¯gu¯,yu,yu−τ−eAt−u¯gu¯,zu,zu−τ2du,J7t=E^sup0≤t≤t1∫0teAt−uhu,yu,yu−τ−eAt−u¯hu,yu,yu−τ2du,J8t=E^sup0≤t≤t1∫0teAt−u¯hu,yu,yu−τ−eAt−u¯hu¯,yu,yu−τ2du,J9t=E^sup0≤t≤t1∫0teAt−u¯hu¯,yu,yu−τ−eAt−u¯hu¯,zu,zu−τ2du.
To estimate (81), we need to estimate Jiti=1,2,3,4,5,7,8,9.
ForJ1t, we have that(83)J1t≤E^sup0≤t≤t1∫0teAt−u−eAt−u¯2fu,yu,yu−τ2du=E^sup0≤t≤t1∫0teAt−u2I−eAu−u¯2fu,yu,yu−τ2du.
Due to the linear growth condition (21) and eAh−I≤AheAT, inequality (83) gives(84)E^sup0≤t≤t1∫0teAt−u−eAt−u¯2fu,yu,yu−τ2du≤G1E^sup0≤t≤t1∫0te2At−uI−eAu−u¯21+yu2+yu−τ2du≤G1e2ATTA2h2e2AT1+2C˜1.
Using (78) and Lipschitz condition (20), it follows from J2t and J3t that(85)J2t≤G2e2ATE^sup0≤t≤t1∫0t1+yu2+yu−τ2u−u¯du≤G2e2ATTh1+2C˜1,(86)J3t≤L1e2|A|TE^sup0≤t≤t1∫0tyu−zu2+yu−τ−zu−τ2du≤2L1e2|A|TT∫0t1E^sup0≤u≤tyu−yu^2dt.
Similar to estimatesJ1t, J2t, and J3t, one obtains(87)J4t≤G1e2ATTA2h2e2AT1+2C˜1,(88)J5t≤G2e2ATTh1+2C˜1,(89)J6t≤2L1e2ATT∫0t1E^sup0≤u≤tyu−yu^|2dt,(90)J7t≤G1e2ATTA2h2e2AT1+2C˜1,(91)J8t≤G2e2ATTh1+2C˜1,(92)J9t≤2L1e2ATT∫0t1E^sup0≤u≤tyu−yu^2dt.
Substituting (84)–(92) into (81) and rearranging (81), we obtain(93)E^sup0≤t≤t1yt−yt^2≤9T+36σ¯2+9Tσ¯4G1e2ATTA2h2e2AT1+2C˜1,(94)+9T+36σ¯2+9Tσ¯4G2e2ATTh1+2C˜1+29T+36σ¯2+9Tσ¯4L1e2ATT∫0t1E^sup0≤u≤tyu−yu^|2dt≤9T+36σ¯2+9Tσ¯4e2ATThG1A2he2AT+G21+2C˜1+29T+36σ¯2+9Tσ¯4L1e2ATT∫0t1E^sup0≤u≤tyu−yu^2dt.
By Gronwall’s inequality, it gives(95)E^sup0≤t≤Tyt−yt^2≤9T+36σ¯2+9Tσ¯4e2AThT+4G1A2he2AT+G21+2C˜1⋅e29T+36σ¯2+9Tσ¯4L1e2|A|TT,and then, letting h⟶0, one can draw the conclusion. The proof is completed.
Theorem2 shows that if the coefficients of f, g, and h obey the Lipschitz condition, in addition to the conditions imposed in Lemma 3, then the exponential Euler approximate solution converges to the exact solution for system (7).


## 4.2. The Mean-Square Stability

In this section, we are in a position to explore the exponential stability of the exact solution and the numerical approximation. For this purpose, we further assume thatft,0,0=0, gt,0,0=0, and ht,0,0=0. Therefore, system (7) admits a trivial solution. Moreover, from condition (20), it is easy to get(96)ft,x,y2∨gt,x,y2∨ht,x,y2≤L1x2+y2,∀x,y∈Rd.In the following, we first give some necessary assumptions and definitions for the mean-square exponential stability.Definition 7.
Equation (7) is said to be mean-square exponentially stable if there exist positive constants v and C, which is dependent on the initial data y0t∈Cℱ0b0,T;Rd and independent of t, such that(97)E^yt2≤Ce−vt,t≥0.
For convenience, we introduce the logarithmic normμA proposed in [38].Definition 8.
The logarithmic normμA of A is defined by(98)μA=limΔ⟶0+I+ΔA−1Δ.
In particular, if⋅ is an inner product norm, μA can also be written as(99)μA=maxy≠0Ay,yy2.Theorem 3.
Assume thatf, g, and h satisfy the Lipschitz condition (20). If there exist positive constants λ1 and λ2, such that(100)2xTft,x,y≤−λ1x2+λ2y2,x,y∈Rd,holds, then the analytical solution of equation (7) is mean-square exponentially stable under the condition(101)λ1−2μA−λ2−6σ¯4L1−4σ¯4L1>0.Proof.
LetVt,yt=1+yt2=1+yTtyt. By the matrix derivative rule and the Ito^ formula, we can derive that(102)dVt,yt=Vtt,yt+Vyt,ytft,yt,yt−τdt+Vyt,ytgt,yt,yt−τdωt+Vyt,ytht,yt,yt−τ+12tracegTVyyt,yt,yt−τgdωt.
The aforementioned equation is equivalent to the following stochastic integral equation:(103)Vt,yt=V0,y0+∫0tVss,ys+Vys,ysfs,ys,ys−τds+∫0tVys,ysgs,ys,ys−τdωs+∫0tVys,yshs,ys,ys−τ+12tracegTVyys,ys,ys−τgdωs,that is,(104)1+yt2=1+y02+2∫0tys,Ays+fs,ys,ys−τds+2∫0tys,gs,ys,ys−τdωs+2∫0tys,hs,ys,ys−τdωs+∫0tgs,ys,ys−τ2dωs.
Taking G-expectation on both sides of (101), we get(105)E^1+yt2=1+y02+2E^∫0tys,Ays+fs,ys,ys−τds+2E^∫0tys,gs,ys,ys−τdωs+2E^∫0tys,hs,ys,ys−τdωs+E^∫0tgs,ys,ys−τ2dωs.
With the condition in (100), the inequality in (105) obtains(106)E^sup0≤s≤t1+ys2≤1+y02+E^∫0t2μA−λ1ys2+λ2ys−τ2ds+2E^sup0≤s≤t∫0syu,gu,yu,yu−τdωu+2E^sup0≤s≤t∫0syu,hu,yu,yu−τdωu+E^sup0≤s≤t∫0sgu,yu,yu−τ2dωu.
Combining Proposition3 with Lemma 1, we have(107)2E^sup0≤s≤t∫0syu,gu,yu,yu−τdωu≤2σ¯2E^∫0tys2gs,ys,ys−τ2ds1/2≤2σ¯2E^sup0≤s≤tys2∫0tgs,ys,ys−τ2ds1/2≤12E^sup0≤s≤t1+ys2+2σ¯4L1E^∫0tys2+ys−τ2ds.
Similarly, we have(108)2E^sup0≤s≤t∫0syu,hu,yu,yu−τdωu≤2σ¯4E^∫0styshs,ys,ys−τds≤σ¯4E^∫0t2L14yshs,ys,ys−τL14ds≤σ¯4E^∫0tL1ys2+hs,ys,ys−τ2L1ds≤σ¯4L1E^∫0tys2+ys2+ys−τ2ds≤2σ¯4L1E^∫0tys2+ys−τ2ds.and(109)E^sup0≤s≤t∫0sgu,yu,yu−τ2dωu≤σ¯4L1E^∫0tys2+ys−τ2ds.
Substituting (107)–(109) into (106), we obtain(110)E^sup0≤s≤t1+ys2≤1+y02+E^∫0t2μA−λ1yu2+λ2yu−τ2du+12E^sup0≤s≤t1+ys2+2σ¯4L1E^∫0tys2+ys−τ2ds+2σ¯4L1E^∫0tys2+ys−τ2ds+σ¯4L1E^∫0tys2+ys−τ2ds,that is,(111)E^sup0≤s≤t1+ys2≤21+y02+22μA−λ1+λ2+6σ¯4L1+4σ¯4L1E^∫0tsup0≤s≤tys2ds.
From the Gronwall inequality, we have(112)E^sup0≤s≤tys2≤21+y02e22μA−λ1+λ2+6σ¯4L1+4σ¯4L1t≤Ce−vt.where C=21+y02 and v=2λ1−2μA−λ2−6σ¯4L1−4σ¯4L1. The proof is completed.Then the question is, will the exponential Euler method reproduce the stability of analytical solutions of equation (7) under the Lipschitz condition? In order to answer this question, we introduce the following definition of mean-square stability from [39] at first and present the stability theorem of the exponential Euler approximation.Definition 9.
A numerical method is said to be asymptotically mean-square stable (with respect to a given G-SLSDDE) if(113)limn⟶∞E^yn2=0.Theorem 4.
Let (H1) hold. Assume that there exist positive constantsλ3 and λ4, such that(114)2σ¯2xTht,x,y≤−λ3x2+λ4y2,x,y∈Rd.
If the inequalitiesL1σ¯2−λ1−λ3<0 and 2μA+λ2+λ4+λ1+λ3<0 hold, then the exponential Euler method is asymptotically mean-square stable for all(115)hn<h0=1/2λ1+1/2λ3−1/2L1σ¯2L1+L1σ¯4+L1σ¯2.Proof.
Squaring both sides of (71), we have(116)yn+12=e2μAhnyn2+f2tn,yn,yn−mhn2+g2tn,yn,yn−mΔωn2+h2tn,yn,yn−mΔωn2+2e2μAhnynTftn,yn,yn−mhn+2e2μAhnynTgtn,yn,yn−mΔωn+2e2μAhnynThtn,yn,yn−mΔωn+2e2μAhnftn,yn,yn−mTgtn,yn,yn−mhnΔωn+2e2μAhnftn,yn,yn−mThtn,yn,yn−mhnΔωn+2e2μAhngtn,yn,yn−mThtn,yn,yn−mΔωnΔωn.
Taking G-expectation on both sides of (116) and considering that yk and yk−m are ℱtk-measurable, by Proposition 3, one can get(117)E^g2tk,yk,yk−mΔωk2−σ¯2hk=E^E^g2tk,yk,yk−mΔωk2−σ¯2hk|ℱtk=E^g2tk,yk,yk−mE^Δωk2−σ¯2hk|ℱtk=0,E^ykTgtk,yk,yk−mΔωk=E^E^ykTgtk,yk,yk−mΔωk|ℱtk=E^ykTgtk,yk,yk−mE^Δωk|ℱtk=0.
By Proposition2, one obtains(118)E^h2tk,yk,yk−mΔωk2−σ¯4hk2=E^E^h2tk,yk,yk−mΔωk2−σ¯4hk2|ℱtk=E^g2tk,yk,yk−mE^Δωk2−σ¯4hk2|ℱtk≤0,E^ykThtk,yk,yk−mΔωk=E^E^ykTgtk,yk,yk−mΔωk|ℱtk=E^ykThtk,yk,yk−mE^Δωk|ℱtk≤σ¯2hkE^ykThtk,yk,yk−m,E^gTtk,yk,yk−mhtk,yk,yk−mΔωkΔωk=E^E^gTtk,yk,yk−mhtk,yk,yk−mΔωkΔωk|ℱtk=E^gTtk,yk,yk−mhtk,yk,yk−mE^ΔωkΔωk|ℱtk≤σ¯2hkE^gTtk,yk,yk−mhtk,yk,yk−mE^Δωk|ℱtk=0,E^fTtk,yk,yk−mhtk,yk,yk−mhkΔωk=E^E^fTtk,yk,yk−mhtk,yk,yk−mhkΔωk|ℱtk=hkE^fTtk,yk,yk−mhtk,yk,yk−mE^Δωk|ℱtk≤σ¯2hk2E^fTtk,yk,yk−mhtk,yk,yk−m.
Then, it follows from (100), (114), and (99) that(119)E^yn+12=e2μAhn|E^yn2+E^f2tn,yn,yn−mhn2+E^g2tn,yn,yn−mσ¯2hn+E^h2tn,yn,yn−mσ¯4hn2+2ynTftn,yn,yn−mhn+2σ¯2hnE^ynThtn,yn,yn−m+2σ¯2hn2E^ftn,yn,yn−mThtn,yn,yn−m|≤e2μAhn|E^yn2+E^L1yn2+L1yn−m2hn2+E^L1yn2+L1yn−m2σ¯2hn+E^L1yn2+L1yn−m2σ¯4hn2+−λ1E^yn2+λ2E^yn−m2hn+hnE^−λ3yn2+λ4yn−m2+2σ¯2hn2E^L1yn2+L1yn−m2|=e2μAhn|1+L1hn2+L1σ¯2hn+L1σ¯4hn2−λ1hn−λ3hn+L1σ¯2hn2E^yn2+L1hn2+L1σ¯2hn+L1σ¯4hn2+λ2hn+λ4hn+L1σ¯2hn2E^yn−m2=A1E^yn2+A2E^yn−m2,where A1=e2μAhn1+L1σ¯2−λ1−λ3hn+L1+L1σ¯4+L1σ¯2hn2 and A2=e2μAhnL1σ¯2+λ2+λ4hn+L1+L1σ¯4+L1σ¯2hn2.
ConsideringL1σ¯2−λ1−λ3<0, for any hn<h0=1/2λ1+1/2λ3−1/2L1σ¯2/L1+L1σ¯4+L1σ¯2, the following inequalities hold:(120)A1≤e2μAhn1+12L1σ¯2−12λ1−12λ3hn,(121)A1+A2≤A1+A2=e2μAhn1+12L1σ¯2−12λ1−12λ3hn+e2μAhnL1σ¯2+λ2+λ4hn+L1+L1σ¯4+L1σ¯2hn2≤e2μAhn1+12λ1+12λ3−12L1σ¯2hn+e2μAhnL1σ¯2+λ2+λ4+12λ1+12λ3−12L1σ¯2hn≤e2μAhn1+λ2+λ4+λ1+λ3hn.
Therefore,(122)E^yn+12≤e2μAhn1+λ2+λ4+λ1+λ3hnmaxE^yn2,E^yn−m2≤e2μA+λ2+λ4+λ1+λ3hnmaxE^yn2,E^yn−m2.
The condition2μA+λ2+λ4+λ1+λ3<0 gives that e2μA+λ2+λ4+λ1+λ3hn<1; it is not difficult to see that E^yn2≤E^yn−m2; therefore,(123)E^yn+12≤e2μA+λ2+λ4+λ1+λ3hnE^yn−m2≤e22μA+λ2+λ4+λ1+λ3hnE^yn−2m2⋮≤en−2/m2μA+λ2+λ4+λ1+λ3hnE^y02,and hence(124)limn⟶∞E^yn2=0.
The above theorem gives the sufficient conditions for keeping mean-square stability by exponential Euler method for equation (7). On the premise of its stability, it is found that the stability and step size of exponential Euler method depend on the norm of A.


## 5. Numerical Examples

In this section, we discuss a numerical example to illustrate the effectiveness of the obtained results.Example 1.
Letωt be a scalar G-Brownian motion with ωt∼N0,1/8,1/4. Consider the scalar nonlinear G-SLSDDE with the form(125)dyt=−12yt−18yt−sinyt−1dt+12yt−1dωt−14ytdωt,t≥0,yt=t+1,t∈−1,0.
Then, the numerical solution of this system is convergent and asymptotically mean-square stable.
Obviously, from system (125), one can obtain μA=−1/2 and ft,0,0=gt,0,0=ht,0,0=0. It is easy to see that f,g,h satisfy the condition in (H1). Moreover, ft,x,y2≤1/32x2+y2, gt,x,y2=1/4y2, and ht,x,y2=1/16x2. Therefore, ft,x,y2∨gt,x,y2∨ht,x,y2≤1/4x2+y2; that is, L1=1/4.
Further, by Young’s inequality, one can obtain(126)2xTft,x,y=2xT−18x−siny≤−316x2+14y2,2σ¯2xTht,x,y=214xT−14x≤−18x2+18y2.
Therefore,λ1=3/16, λ2=1/4, λ3=1/8, and λ4=1/8 satisfy L1σ¯2−λ1−λ3<0 , 2μA+λ2+λ4+λ1+λ3<0, and λ1−2μA−λ2−6σ¯4L1−4σ¯4L1>0. Then, it follows from Theorem 3 that the trivial solution of the example is mean-square exponentially stable.
Noticing that(127)h0=1/2λ1+1/2λ3−1/2L1σ¯2L1+L1σ¯4+L1σ¯2=821,by Theorem 2 and Theorem 4, we see that the exponential Euler method is convergent and asymptotically mean-square stable.
On the other hand, by the exponential Euler method, choose the step sizeh<8/21 and the initial value y0=1 to simulate the numerical solution yn and E^yn2 for system (125), which are shown in Figures 1 and 2 , respectively. It follows from Figure 1 that the numerical solution of system (125) converges to zero. Moreover, it follows from Figure 2 that numerical solution of system (125) is asymptotically mean-square stable. In brief, the numerical simulation results are consistent with our theoretical results.Figure 1
The numerical solution withh=0.1 of the exponential Euler method.Figure 2
The G-expectation of numerical solution withh=0.1 of the exponential Euler method.


## 6. Conclusions

This paper is devoted to applying the exponential integrators to the semilinear stochastic delay differential equations driven by G-Brownian motion (G-SLSDDEs) and dealing with the convergence and stability properties of exponential integrators for G-SLSDDEs. It first investigates some suitable conditions for the mean-square stability of the analytic solution and then shows that the exponential integrators numerical solution converges to the analytic solution for the G-SLSDDEs with the strong order 1/2. Furthermore, it proves that the exponential Euler method can keep the mean-square exponential stability of the analytic solution under some restrictions on the step size.G-framework is a new study area which many scholars begin to pay attention to. Recently, some results related to G-framework have been obtained; however, it is a pity that there are few numerical conclusions. In the future, we will further discover more efficient numerical methods, such as the transferred Legendre pseudospectral method in [40], to solve semilinear stochastic delay differential equations with time-variable delay driven by G-Brownian motion. Influenced by experience gained from solving stochastic fractional differential equations driven by fractional Brownian motion, we will further study stochastic fractional differential equations driven by G-Brownian motion with “dy (t)” as fractional order in future research [41].



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