# On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations

Authors: Haiyan Yuan; Jihong Shen; Cheng Song
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method withθ∈(1/2,1] is asymptotically mean square stable for all positive step sizes, and the split-step theta method with θ∈[0,1/2] is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.


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## Body

## 1. Introduction

Stochastic functional differential equations (SFDEs) play important roles in science and engineering applications, especially for systems whose evolutions in time are influenced by random forces as well as their history information. When the time delays in SFDEs are constants, they turn into stochastic delay differential equations (SDDEs). Both the theory and numerical methods for SDDEs have been well developed in the recent decades; see [1–8]. Recently, many dynamical systems not only depend on the present and the past states but also involve derivatives with delays; they are described as the neutral stochastic delay differential equations (NSDDEs). Compared to the stochastic differential equations and the stochastic delay differential equations, the study of the neutral stochastic delay differential equations has just started. In 1981, Kolmanovskii and Myshkis [9] took the environmental disturbances into account, introduced the neutral stochastic delay differential equations (NSDDEs), and gave their applications in chemical engineering and aeroelasticity. The analytical solutions of NSDDEs are hard to obtain; many authors have to study the numerical methods for NSDDEs. Wu and Mao [10] studied the convergence of the Euler-Maruyama method for neutral stochastic functional differential equations under the one-side Lipschitz conditions and the linear growth conditions. In 2009, Zhou and Wu [11] studied the convergence of the Euler-Maruyama method for NSDDEs with Markov switching under the one-side Lipschitz conditions and the linear growth conditions. The convergence of θ-method and the mean square asymptotic stability of the semi-implicit Euler method for NSDDEs were studied by Gan et al. [12], Zhou and Fang [13], and Yin and Ma [14], respectively. Later, the almost sure exponential stability of Euler-Maruyama method for NSDDEs was studied in [15] with the discrete semimartingale convergence theorem.To the best of our knowledge, most of these studies have focused on the convergence of numerical solutions for NSDDEs; the stability and dissipativity of numerical solutions for them are rarely concerned.The aim of this paper is to study the mean square stability and dissipativity of the split-step theta method with some conditions and the step constrained for NSDDEs.The paper is organized as follows. In Section2, some stability definitions about the analytic solutions for NSDDEs are introduced; some notations and preliminaries are also presented in this section. In Section 3, the split-step theta method is introduced and used to solve the NSDDEs; the asymptotic stability of the split-step theta method is proved. In Section 4, the long time behavior of numerical solution is studied and the mean square dissipativity result of the method is illustrated. In Section 5, some numerical experiments are given to confirm the theoretical results.


## 2. Exponential Mean Square Stability of Analytic Solution

Let· denote both the Euclidean norm in Rd and the trace (or Frobenius) norm in Rd×l (denoted by A=trace(AΤA)); if A is a vector or matrix, its transpose is denoted by AΤ. Let Ω,F,Ftt≥0,P define a complete probability space with a filtration Ftt≥0 which is increasing and right continuous, and F0 contain all P-null sets. Let w(t)=(w1(t),w2(t),…,wl(t))Τ denote standard l-dimensional Brownian motion on the probability space. In this paper we talk about the d-dimensional NSDDEs with the following form:(1)dyt-Nyt-τ=ft,yt,yt-τdt+gt,yt,yt-τdwt,t≥0,yt=φt,t∈-τ,0,where N:Rd↦Rd, f:R+×Rd×Rd↦Rd, and g:R+×Rd×Rd↦Rd×l are the Borel measurable functions. τ is a positive constant delay, and φ(t) is F0-measurable, C(-τ,0;Rd)-valued random variable which satisfies(2)sup-τ≤t≤0ΕφΤtφt<+∞with the notation E denoting the mathematical expectation with respect to P.The following conditions (a1) and (a2) are standard for the existence and uniqueness of the solution for (1).(a1) The Local Lipschitz Condition. There exist constants KL>0 and L>0 such that(3)ft,x1,y1-ft,x2,y22∨gt,x1,y1-gt,x2,y22≤KLx1-x22+y1-y22, for all x1∨x2∨y1∨y2≤L and t∈R+, where a∨b represents max⁡a,b and a∧b represents min⁡a,b.(a2) The Linear Growth Condition. There exists a constant KG>0, such that(4)ft,x,y2∨gt,x,y2∨Nx2≤KG1+x2+y2, for all (t,x,y)∈R+×R×R.As an especial case of Theorem 3.1 in Mao’s monograph (see [6]), we can easily know that under hypothesis (a1) and (a2), system (1) has a global unique continuous solution on t≥-τ, which is denoted by y(t).Now we recall some stability concepts for the solution of (1).Definition 1 (see [6]).
The trivial solution of (1) is said to be exponentially mean square stable, if there exists a pair of constants r>0 and C>0, such that, whenever sup-τ≤t≤0ΕφΤtφt<+∞,(5)ΕyΤtyt≤Csup-τ≤t≤0ΕφΤtφte-rt,t≥0.Lemma 2.
Assume that there exist a symmetric, positive definited×d matrix Q and positive constants μ1,μ2, and λ∈(0,1) such that for all (t,x,y)∈R+×Rd×Rd(6)Nx≤λx,(7)x-NyΤQft,x,y+12tracegΤt,x,yQgt,x,y≤-μ1xΤQx+μ2yΤQy.If conditions(8)0<λ<12,μ1>μ21-2λ2hold, then the trivial solution of (1) is exponentially mean square stable.Remark 3.
In general, we requireλ≠0. When λ=0, (1) becomes a stochastic delay differential equation. Many stability and dissipativity results have been studied in the literature (see [5, 16]).By Lemma2, the following result can easily be obtained.Theorem 4.
Suppose (6) holds. Assume that there are positive constants λ1,λ2, and K, such that, for all x,y∈Rd,(9)x-NyΤQft,x,y≤-λ1xΤQx+λ2yΤQy,ft,x,y2∨gt,x,y2≤Kx2+y2.If conditions(10)0<λ<12,λ1>12K+2λ2+K21-2λ2hold, then the trivial solution of (1) is exponentially mean square stable.Proof.
Consider (9) and the inequality; we get the following inequality:(11)x-NyΤQft,x,y+12tracegΤt,x,yQgt,x,y≤-λ1xΤQx+λ2yΤQy+12KxΤQx+yΤQy≤-λ1-12KxΤQx+λ2+12Kμ2yΤQy.Let μ1=λ1-(1/2)K, μ2=λ2+(1/2)K; when conditions (10) hold, we get that (12)μ1>μ21-2λ2.Using Lemma 2, we can easily prove that the trivial solution of (1) is exponentially mean square stable.


## 3. The Stability of the Split-Step Theta Method

The split-step theta method is proved to be able to keep the mean square asymptotic stability of the exact solution under the sufficient conditions of the asymptotic stability of the exact solution, so in this paper we use the split-step theta method to solve the NSDDE.Appling the split-step theta (SST) method into problem (1) gives the following form:(13)Yn-NYn-m=yn-Nyn-m+θΔtftn+θΔt,Yn,Y-n,(14)Y-n=Yn-m,(15)yn+1-Nyn+1-m=yn-Nyn-m+Δtftn+θΔt,Yn,Y-n+gtn+θΔt,Yn,Y-nΔwn,where the step size Δt=τ/m, m is an integer, yi is an approximation to y(ti), ti=iΔt, i=1,2,…, and yk=Yk=φ(kΔt) for k=-m,-m+1,…,0. θ∈[0,1] is a fixed parameter, and Δwk≔w((k+1)Δt)-w(kΔt) is the Brownian increment.Whenθ=0 the split-step theta method is simplified into the split-step forward Euler method and when θ=1 the split-step theta method is simplified into the split-step backward Euler method. They were discussed for stochastic differential equations in [17–20]. In order to consider the stability property of scheme (13)–(15) we should give some stability concepts for numerical methods firstly.Definition 5 (see [16]).
For a given step sizeΔt, a numerical method is said to be exponentially mean square stable if there is a pair of positive constants γ and C such that for any initial data φ(t) the numerical solution yn produced by the method satisfies(16)ΕynΤyn≤Ce-γtnsup-τ≤t≤0ΕφΤtφt,∀n≥0.Definition 6 (see [16]).
For a given step sizeΔt, a numerical method is said to be asymptotically mean square stable if for any initial data φ(t) the numerical solution yn produced by the method satisfies(17)limn→∞⁡ΕynΤyn=0.Theorem 7.
Assume that system (1) satisfies (7) with -μ1+μ2<0; then the SST method (13)–(15) with θ∈(1/2,1] is asymptotically mean square stable for all Δt>0. If we further assume that there exist constants K1 and K2 such that (18)fΤt,x,yQft,x,y≤K1xΤQx+K2yΤQy,t,x,y∈R+×Rd×Rd,then, for any θ∈[0,1/2), there exists a constant Δt0 depending on θ such that the method is asymptotically mean square stable for Δt∈(0,Δt0).Proof.
From (15) it follows that(19)yn+1-Nyn+1-mΤQyn+1-Nyn+1-m=yn-Nyn-mΤQyn-Nyn-m+Δt2fΤtn+θΔt,Yn,Y-nQftn+θΔt,Yn,Y-n+ΔwnΤgΤtn+θΔt,Yn,Y-nQgtn+θΔt,Yn,Y-nΔwn+2yn-Nyn-mΤΔtQftn+θΔt,Yn,Y-n+2yn-Nyn-mΤQgtn+θΔt,Yn,Y-nΔwn+2ΔtfΤtn+θΔt,Yn,Y-nQgtn+θΔt,Yn,Y-nΔwn.Since w(t)=w1t,w2t,…,wltΤ is a standard l-dimensional Brownian motion we have that (20)ΕΔwi=0,ΕΔwi2=Δt,ΕΔwnΤgΤtn+θΔt,Yn,Y-nQgtn+θΔt,Yn,Y-nΔwn=ΔtΕtracegΤtn+θΔt,Yn,Y-nQgtn+θΔt,Yn,Y-n.Let xn=yn-Nyn-m, Xn=Yn-NYn-m, n=0,1,…, substitute the designation into (19) and then, taking expectation on both sides, one receives(21)Εxn+1ΤQxn+1≤ΕxnΤQxn+1-2θΔt2fΤtn+θΔt,Yn,Y-nQftn+θΔt,Yn,Y-n+2ΔtΕYn-NYn-mΤΔtQftn+θΔt,Yn,Y-n+ΔtΕtracegΤtn+θΔt,Yn,Y-nQgtn+θΔt,Yn,Y-n,which, combined with (7), gives(22)Εxn+1ΤQxn+1≤ΕxnΤQxn+2ΔtΕ-μ1YnΤQYn+μ2Y-nΤQY-n+1-2θΔt2fΤtn+θΔt,Yn,Y-nQftn+θΔt,Yn,Y-n.In the case of θ>1/2, using(23)Δtftn+θΔt,Yn,Y-n=1θXn-xn,2XnΤQxn≤2θ-1--μ1+μ2Δtθ22θ-1XnΤQXn+2θ-12θ-1--μ1+μ2Δtθ2xnΤQxn,then we have (24)Εxn+1ΤQxn+1≤1+-μ1+μ2Δt2θ-12θ-1--μ1+μ2Δtθ2ΕxnΤQxn-2Δtμ2ΕYnΤQYn+2Δt1-λ2μ2+λ2μ1ΕY-nΤQY-n.Let (25)k=max1+-μ1+μ2Δt2θ-12θ-1--μ1+μ2Δtθ2,μ21-λ2μ2+λ2μ11/m;we can deduce that 0<k<1.
By induction, the following results are obtained from (24):(26)Εxn+1ΤQxn+1≤kn+1Εx0ΤQx0-2Δtμ2∑j=0nkn-jΕYjΤQYj+2Δt1-λ2μ2+λ2μ1∑j=0nkn-jΕY-jΤQY-j.Using condition (14), we can get the following inequality:(27)∑j=0nkn-jΕY-jΤQY-j≤mkn-m+1max-m≤j≤-1⁡ΕYjΤQYj+k-m∑j=0n-m+1kn-jΕYjΤQYj.Therefore,(28)Εxn+1ΤQxn+1≤kn+1Εx0ΤQx0+2τ1-λ2μ2+λ2μ1k-mmax-m≤j≤-1⁡ΕYjΤQYj-2Δtμ2-1-λ2μ2+λ2μ1k-m∑j=0n-m+1kn-jΕYjΤQYj.It can be deduced from (28) and (25) that -μ2-1-λ2μ2+λ2μ1k-m≤0, so, we can have the following inequality:(29)Εxn+1ΤQxn+1≤kn+1Εx0ΤQx0+2τ1-λ2μ2+λ2μ1k-mmax-m≤j≤-1⁡ΕYjΤQYj.On the other hand, we know that(30)yn+1=yn+1-Nyn+1-m+Nyn+1-m≤xn+1+Nyn+1-m,then we get(31)Εyn+1ΤQyn+1≤2Εxn+1ΤQxn+1+2λ2Εyn+1-mΤQyn+1-m;define(32)ε0=kn+1Εx0ΤQx0+2τ1-λ2μ2+λ2μ1k-mmax-m≤j≤-1⁡ΕYjΤQYj;the following inequality could be deduced from (31):(33)Εyn+1ΤQyn+1≤21-2λ2ε0+2λ2n/m+1max-m≤j≤-1⁡ΕyjΤQyj,which implies that the method is asymptotically mean square stable.
For the case thatθ∈[0,1/2), with the hypothesis (18) and (22) we can obtain the following inequality: (34)Εxn+1ΤQxn+1≤ΕxnΤQxn+Δt1-2θΔtK1-2μ1ΕYnΤQYn+Δt1-2θΔtK2+2μ2ΕY-nΤQY-.A combination of (13) and (18) gives(35)xnΤQxn≤L1YnΤQYn+L2Y-nΤQY-n,where L1=(1+θΔt)(2+θΔtK1), L2=(1+θΔt)(2λ2+θΔtK2).
Let(36)Δt0=+∞,θ=12,-2-μ1+μ21-2θK1+K2,θ∈0,12;then, for any fixed Δt∈(0,Δt0), 2(-μ1+μ2)+Δt(1-2θ)(K1+K2)<0, there exists a small positive number ε such that(37)2-μ1+μ2+Δt1-2θK1+K2+L1+L2Δtε<0.Therefore,(38)Εxn+1ΤQxn+1≤1-εΕxnΤQxn+Δt1-2θΔtK1-2μ1+L1ΔtεΕYnΤQYn+Δt1-2θΔtK2+2μ2+L2ΔtεΕY-nΤQY-n.Let k~=max⁡1-ε,1-2θΔtK2+2μ2+L2/Δtε/-1-2θΔtK1-2μ1+L1/Δtε1/m; then 0<k~<1. Similar to the derivation of the first part, the following inequality can be proved from (38): (39)Εxn+1ΤQxn+1≤kn+1Εx0ΤQx0+L~k~-mmax-m≤j≤-1⁡ΕYjΤQYj,where L~=τ(1-2θΔtK2+2μ2+L2/Δtε).
Similar to the proof of (31), we can prove that when Δt∈(0,Δt0) the method is asymptotically mean square stable; the proof of theorem is completed.Remark 8.
For system (1) with N=0, it becomes a stochastic delay differential equation; the mean square stability of the theta method has been studied in [16]; Theorem 7 can be regarded as an extension of Theorem 3.4 presented in [16].Remark 9.
For the NSDDEs, the mean square asymptotic stability of the BEM method has been studied by Wang and Chen in [15]; it has shown that BEM method can reproduce the mean square stability of the exact solutions; Theorem 7 improves the result in [15].


## 4. Mean Square Dissipativity

The numerical solutions’ long time dynamic behavior will be studied in this section. Before it, we make the following hypothesis: assume that there exist a symmetric, positive definited×d matrix Q and positive constants μ1, μ2, and γ such that, for all (t,x,y)∈R+×Rd×Rd, the following inequality exists:(40)x-NyΤQft,x,y+12tracegΤt,x,yQgt,x,y≤γ-μ1xΤQx+μ2yΤQy.Now we state and prove some conclusions.Definition 10 (see [16]).
Assume that system (1) satisfies (40). The numerical method is said to be dissipative if when the method is applied to problem (1) with constraint τ=mh, there exists a constant C such that, for any initial values, there exists n0, depending only on initial values φ(t), such that(41)ΕynΤQyn≤C,n≥n0.Theorem 11.
Assume that system (1) satisfies (40); there exists a constant C such that, for any initial values, there exists n0 depending only on the initial values φ(t), when n≥n0, the numerical solution yn generated by the SST method (13)–(15) with θ∈(1/2,1], such that(42)ΕynΤQyn≤C.Proof.
Consider (21) and (40); the following inequality can be obtained:(43)Εxn+1ΤQxn+1≤ΕxnΤQxn+2Δtγ+2ΔtΕ-μ1YnΤQYn+μ2Y-nΤQY-n+1-2θΔt2fΤtn+θΔt,Yn,Y-nQftn+θΔt,Yn,Y-n.We can get the following inequality the same as the derivation of (28): (44)Εxn+1ΤQxn+1≤kn+1Εx0ΤQx0+2τ1-λ2μ2+λ2μ1k-mmax-m≤j≤-1⁡ΕYjΤQYj-2Δtμ2-1-λ2μ2+λ2μ1k-m∑j=0n-m+1kn-jΕYjΤQYj+2Δtγ∑j=0nkj,where 0<k<1 is the same as defined in (25).
Because-μ2-1-λ2μ2+λ2μ1k-m≤0, we have the following inequality:(45)Εxn+1ΤQxn+1≤kn+1Εx0ΤQx0+2τ1-λ2μ2+λ2μ1k-mmax-m≤j≤-1⁡ΕYjΤQYj+2γΔt1-k.Let (46)ε1=kn+1Εx0ΤQx0+2τ1-λ2μ2+λ2μ1k-mmax-m≤j≤-1⁡ΕYjΤQYj+2γΔt1-k;the following inequality could be deduced from (31):(47)Εyn+1ΤQyn+1≤2Εxn+1ΤQxn+1+2λ2Εyn+1-mΤQyn+1-m≤2ε1+2λ2Εyn+1-mΤQyn+1-m≤C,where C=2ε1/1-2λ2+ε. The theorem is completed.Theorem11 means that the discrete system possesses a bounded absorbing set in the sense of mean square. The numerical solution trajectory from any initial date will enter the set in a finite time and thereafter remain inside. It is called mean square dissipativity.Remark 12.
For the study of the dissipativity of numerical methods for deterministic delay differential equations with constant delays, Huang and Chang studied the dissipativity of Runge-Kutta methods and multistep Runge-Kutta methods in [21, 22].


## 5. The Numerical Experiment

In this section, we will give a numerical experiment to illustrate the stability and dissipativity result obtained in Sections3 and 4. Consider the following nonlinear scalar neutral stochastic delay differential equation:(48)dyt-0.25sin⁡yt-1=-8yt+sin⁡yt-1dt+yt-1dWt,t≥0,yt=t+1,-1≤t≤0.It is easy to verify that nonlinear neutral stochastic delay differential equation (48) satisfies the conditions of Theorem 7; the corresponding parameters are given as follows: (49)
λ
=
1
4
,
μ
1
=
8
,
μ
2
=
1
2
,
K
1
=
64
,
K
2
=
1
,
Δ
t
0
=
+
∞
,
θ
=
1
2
0.2884
,
θ
∈
0
,
1
2
.The initial condition is given byy(t)=t+1, t∈[-1,0], where we take τ=1. In the following tests, we show the influence of step size Δt and the parameter θ on M-S stability of the SST method; the data used in all figures are obtained by the mean square of data by 200 trajectories; that is, Εyn2≈1/200)∑i=1200[yn(i)]2, Εyn≈1/200)∑i=1200yn(i), where yn(i) denotes the numerical solution of y(tn) in the ith trajectory.Taking step sizesΔt=0.1, Δt=0.2, Δt=0.3, and Δt=0.6, we obtain the numerical solutions of (48), and the numerical solutions are displayed in Figures 1–6, respectively. We can see that when θ=0.6, the SST method is asymptotically mean square stable for all the step sizes selected, but when θ=0.1, the SST method is asymptotically mean square stable only for the step sizes Δt≤0.2884; it is not mean square stable for the step sizes Δt>0.2884.Figure 1
Mean square stability of SST method withθ=0.6 and Δt=0.1.Figure 2
Mean square stability of SST method withθ=0.6 and Δt=0.6.Figure 3
Mean square stability of SST method withθ=0.1 and Δt=0.1.Figure 4
Mean square stability of SST method withθ=0.1 and Δt=0.2.Figure 5
Unstable test for SST method withθ=0.1 and Δt=0.3.Figure 6
Unstable test for SST method withθ=0.1 and Δt=0.6.



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