We investigate the almost surely asymptotic stability of Euler-type
methods for neutral stochastic delay differential equations (NSDDEs) using the discrete semimartingale
convergence theorem. It is shown that the Euler method and the backward Euler method can reproduce the almost surely asymptotic stability of exact solutions to NSDDEs under additional conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.

1. Introduction

The neutral stochastic delay differential equation (NSDDE) has attracted much more attention, and much work (see [1–4]) has been done. For example, Mao [2] studied the existence and uniqueness, moment and pathwise estimates, and the exponential stability of the solution to the NSDDE. Moreover, Mao et al. [4] studied the almost surely asymptotic stability of the NEDDE with Markovian switching:d[x(t)-N(x(t-τ),r(t))]=f(t,x(t),x(t-τ),r(t))dt+g(t,x(t),x(t-τ),r(t))dB(t).

Since most NSDDEs cannot be solved explicitly, numerical solutions have become an important issue in the study of NSDDEs. Convergence analysis of numerical methods for NSDDEs can be found in [5–7]. On the other hand, stability theory of numerical solutions is one of the fundamental research topics in the numerical analysis. For stochastic differential equations (SDEs) as well as stochastic delay differential equations (SDDEs), moment stability and asymptotic stability of numerical solutions have received much more attention (e.g., [8–13] for moment stability and [12–14] for asymptotic stability). Recently, Wang and Chen [15] studied the mean-square stability of the semi-implicit Euler method for NSDDEs. We aim in this paper to study the almost surely asymptotic stability of Euler-type methods for NSDDEs using the discrete semimartingale convergence theorem. The discrete semimartingale convergence theorem (cf. [16, 17]) plays an important role in the almost surely asymptotic stability analysis of numerical solutions to SDEs and SDDEs [17–19]. Using the discrete semimartingale convergence theorem, we show that Euler-type methods for NSDDEs can preserve the almost surely asymptotic stability of exact solutions under additional conditions.

In Section 2, we introduce some necessary notations and state the discrete semimartingale convergence theorem as a lemma. In Section 3, we study the almost surely asymptotic stability of exact solutions to NSDDEs. Section 4 gives the almost surely asymptotic stability of the Euler method. In Section 5, we discuss the almost surely asymptotic stability of the backward Euler method. Numerical experiments are presented in Section 6.

2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations. Let (Ω,ℱ,{ℱt}t≥0,P) be a complete probability space with filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets). B(t) is a scalar Brownian motion defined on the probability space. |·| denotes the Euclidean norm in Rn. The inner product of x,y in Rn is denoted by 〈x,y〉 or xTy. If A is a vector or matrix, its transpose is denoted by AT. If A is a matrix, its trace norm is denoted by |A|=trace(ATA). Let τ>0 and C([-τ,0];Rn) denote the family of all continuous Rn-valued functions on [-τ,0]. Let Cℱ0b([-τ,0];Rn) be the family of all ℱ0-measurable bounded C([-τ,0];Rn)-valued random variables ξ={ξ(θ):-τ≤θ≤0}.

Consider an n-dimensional NSDDEd[x(t)-N(x(t-τ))]=f(t,x(t),x(t-τ))dt+g(t,x(t),x(t-τ))dB(t),
on t≥0 with initial data {x(θ):-τ≤θ≤0}=ξ∈Cℱ0b([-τ,0];Rn). Here N:Rn→Rn, f:R+×Rn×Rn→Rn, and g:R+×Rn×Rn→Rn.

Let C1,2(R+×Rn;R+) denote the family of all nonnegative functions V(t,x) on R+×Rn which are continuously once differentiable in t and twice differentiable in x. For each V∈C1,2(R+×Rn;R+), define an operator LV from R+×Rn×Rn to R by LV(t,x,y)=Vt(t,x-N(y))+Vx(t,x-N(y))f(t,x,y)+12trace[gT(t,x,y)Vxx(t,x-N(y))g(t,x,y)],
where Vt(t,x)=∂V(t,x)∂t,Vx(t,x)=(∂V(t,x)∂x1,…,∂V(t,x)∂xn),Vxx(t,x)=(∂2V(t,x)∂xi∂xj)n×n.

As a standing hypothesis, we impose the following assumption on the coefficients N,f, and g.

Assumption 2.1.

Assume that both f and g satisfy the local Lipschitz condition. That is, for each integer i>0, there exists a positive constant Ki such that
|f(t,x,y)-f(t,x¯,y¯)|2∨|g(t,x,y)-g(t,x¯,y¯)|2≤Ki(|x-x¯|2+|y-y¯|2)
for x,y,x¯,y¯∈Rn with |x|∨|x¯|∨|y|∨|y¯|≤i and t∈R+. Assume also that there is a constant κ∈(0,1) such that
|N(x)-N(y)|≤κ|x-y|,∀x,y∈Rn.
Assume moreover that for all t∈R+,
N(0)=0,f(t,0,0)=0,g(t,0,0)=0.

The following discrete semimartingale convergence theorem (cf. [16, 17]) will play an important role in this paper.

Lemma 2.2.

Let {Ai} and {Ui} be two sequences of nonnegative random variables such that both Ai and Ui are ℱi-measurable for i=1,2,…, and A0=U0=0 a.s. Let Mi be a real-valued local martingale with M0=0 a.s. Let ζ be a nonnegative ℱ0-measurable random variable. Assume that {Xi} is a nonnegative semimartingale with the Doob-Mayer decomposition
Xi=ζ+Ai-Ui+Mi.
If limi→∞Ai<∞ a.s., then for almost all ω∈Ωlimi→∞Xi<∞,limi→∞Ui<∞,
that is, both Xi and Ui converge to finite random variables.

3. Almost Surely Asymptotic Stability of the Exact Solution

In this section, we will study the almost surely asymptotic stability of exact solutions to (2.1). To be precise, let us give the definition on the almost surely asymptotic stability of exact solutions.

Definition 3.1.

The solution x(t) to (2.1) is said to be almost surely asymptotically stable if
limt→∞x(t)=0a.s.
for any initial data ξ∈Cℱ0b([-τ,0];Rn).

Theorem 3.2.

Let Assumption 2.1 hold. Assume that there are four positive constants λ1-λ4 such that
2(x-N(y))Tf(t,x,y)≤-λ1|x|2+λ2|y|2,|g(t,x,y)|2≤λ3|x|2+λ4|y|2
for t≥0 and x,y∈Rn. If
λ1-λ3>λ2+λ4,
then, for any initial data ξ∈Cℱ0b([-τ,0];Rn), there exists a unique global solution x(t) to (2.1) and the solution x(t) is almost surely asymptotically stable.

Proof.

Let U(t,x)=|x|2. Using (3.2) and (3.3), we have
LU(t,x,y)=2(x-N(y))Tf(t,x,y)+|g(t,x,y)|2≤-(λ1-λ3)|x|2+(λ2+λ4)|y|2≤-(λ1-λ3)|x|2+(λ1-λ3)|y|2.
Then, from Theorem 3.1 in [4], we conclude that there exists a unique global solution x(t) to (2.1) for any initial data ξ∈Cℱ0b([-τ,0];Rn). According to (3.3), there is a constant α¯>0 such that
λ1-λ3-2α≥(λ2+λ4+2α)eατ∀0≤α≤α¯.
Let V(t,x)=eαtU(t,x). Here α∈(0,α¯]∩(0,(2/τ)log(1/κ)). Then
LV(t,x,y)=eαt(αU(t,x-N(y)))+LU(t,x)≤eαt[α|x-N(y)|2-(λ1-λ3)|x|2+(λ2+λ4)|y|2]≤eαt[-(λ1-λ3-2α)|x|2+(λ2+λ4+2α)|y|2]≤-W(t,x)+W(t-τ,y),
where W(t,x)=(λ1-λ3-2α)eαt|x|2. By Theorem 4.1 in [4], we can obtain that the solution x(t) is almost surely asymptotically stable. The proof is completed.

Theorem 3.2 gives sufficient conditions of the almost surely asymptotic stability of the NSDDE (2.1). Based on these sufficient conditions, we will investigate the almost surely asymptotic stability of Euler-type methods in the following sections.

4. Stability of the Euler Method

Applying the Euler method (EM) to (2.1) yieldsXk+1-N(Xk+1-m)=Xk-N(Xk-m)+f(kh,Xk,Xk-m)h+g(kh,Xk,Xk-m)ΔBk,k=0,1,2,…,Xk=ξ(kh),k=-m,-m+1,…,0.
Here h=τ/m (m is an positive integer) is the stepsize, and ΔBk=B((k+1)h)-B(kh) represents the Browian motion increment. To be precise, let us introduce the definition on the almost surely asymptotic stability of numerical solutions.

Definition 4.1.

The numerical solution Xk to (2.1) is said to be almost surely asymptotically stable if
limk→∞Xk=0a.s.
for any bounded variables ξ(kh), k=-m,-m+1,…,0.

Theorem 4.2.

Let conditions (3.2)-(3.3) hold. Assume that f satisfies the linear growth condition, namely, there exists a positive constant L such that
|f(t,x,y)|2≤L(|x|2+|y|2).
Then there exists a h0>0 such that if h<h0, then for any given finite-valued ℱ0-measurable random variables ξ(kh), k=-m,-m+1,…,0, the EM approximate solution (4.1) is almost surely asymptotically stable.

Proof.

Let Yk=Xk-N(Xk-m). Then, it follows from (4.1) that
Yk+1=Yk+f(kh,Xk,Xk-m)h+g(kh,Xk,Xk-m)ΔBk,k=0,1,2,….
Squaring both sides of (4.4), we have
|Yk+1|2=|Yk|2+|f(kh,Xk,Xk-m)|2h2+|g(kh,Xk,Xk-m)|2h+2〈Yk,f(kh,Xk,Xk-m)〉h+2〈Yk,g(kh,Xk,Xk-m)〉ΔBk+2h〈f(kh,Xk,Xk-m),g(kh,Xk,Xk-m)〉ΔBk+|g(kh,Xk,Xk-m)|2(ΔBk2-h).
Using (3.2) and (4.3), we can obtain that
|Yk+1|2≤|Yk|2+L(|Xk|2+|Xk-m|2)h2+(λ3|Xk|2+λ4|Xk-m|2)h+(-λ1|Xk|2+λ2|Xk-m|2)h+mk,
where
mk=2〈Yk,g(kh,Xk,Xk-m)〉ΔBk+2h〈f(kh,Xk,Xk-m),g(kh,Xk,Xk-m)〉ΔBk+|g(kh,Xk,Xk-m)|2(ΔBk2-h).
It therefore follows that
|Yk+1|2-|Yk|2≤-(λ1-λ3-Lh)h|Xk|2+(λ2+λ4+Lh)h|Xk-m|2+mk,
which implies that
|Yk|2-|Y0|2≤-(λ1-λ3-Lh)h∑i=0k-1|Xi|2+(λ2+λ4+Lh)h∑i=0k-1|Xi-m|2+∑i=0k-1mi.
Note that
∑i=0k-1|Xi-m|2=∑i=-m-1|Xi|2+∑i=0k-m-1|Xi|2.
Then, we have
|Yk|2+(λ2+λ4+Lh)h∑i=k-mk-1|Xi|2≤|Y0|2+(λ2+λ4+Lh)h∑i=-m-1|Xi|2-(λ1-λ3-λ2-λ4-2Lh)h∑i=0k-1|Xi|2+Mk,
where Mk=∑i=0k-1mi. By [18], Mk is a martingale with M0=0. From (3.3), we obtain that
λ1-λ3-λ2-λ4-2Lh>0,as0<h<h0,
where h0=(λ1-λ3-λ2-λ4)/2L. Hence, from Lemma 2.2, we therefore have
limk→∞∑i=0k-1|Xi|2<∞a.s.,0<h<h0.
Then, we conclude that
limk→∞|Xk|2=0a.s.,0<h<h0.
The proof is completed.

Theorem 4.2 shows that if the coefficient f obeys the linear growth condition, in addition to the conditions imposed in Theorem 3.2, then the EM approximate solution (4.1) reproduces the almost surely asymptotic stability of exact solutions to (2.1) for sufficiently small stepsize.

5. Stability of the Backward Euler Method

Applying the backward Euler method (BEM) to (2.1) yieldsXk+1-N(Xk+1-m)=Xk-N(Xk-m)+f((k+1)h,Xk+1,Xk+1-m)h+g(kh,Xk,Xk-m)ΔBk,k=0,1,2,…,Xk=ξ(kh),k=-m,-m+1,…,0.
As a standing hypothesis, we assume that the BEM (5.1) is well defined. The following theorem shows that if the above assumption and the conditions imposed in Theorem 3.2 hold, then the BEM approximate solution (5.1) inherits the almost surely asymptotic stability of exact solutions to (2.1) without any stepsize restriction.

Theorem 5.1.

Let conditions (3.2)-(3.3) hold. Then for any given finite-valued ℱ0-measurable random variables ξ(kh), k=-m,-m+1,…,0, the BEM approximate solution (5.1) is almost surely asymptotically stable.

Proof.

Let Yk=Xk-N(Xk-m). Then, it follows from (5.1) that
Yk+1-f((k+1)h,Xk+1,Xk+1-m)h=Yk+g(kh,Xk,Xk-m)ΔBk,k=0,1,2,….
Squaring both sides of (5.2), we have
|Yk+1|2+|f((k+1)h,Xk+1,Xk+1-m)|2h2=|Yk|2+2〈Yk+1,f((k+1)h,Xk+1,Xk+1-m)〉h+|g(kh,Xk,Xk-m)|2h+2〈Yk,g(kh,Xk,Xk-m)〉ΔBk+|g(kh,Xk,Xk-m)|2(ΔBk2-h).
Using (3.2), we can obtain that
|Yk+1|2≤|Yk|2+(-λ1|Xk+1|2+λ2|Xk+1-m|2)h+(λ3|Xk|2+λ4|Xk-m|2)h+mk,
where
mk=2〈Yk,g(kh,Xk,Xk-m)〉ΔBk+|g(kh,Xk,Xk-m)|2(ΔBk2-h).
It therefore follows that
|Yk+1|2-|Yk|2≤-λ1h|Xk+1|2+λ3h|Xk|2+λ2h|Xk+1-m|2+λ4h|Xk-m|2+mk,
which implies that
|Yk|2-|Y0|2≤-λ1h∑i=0k-1|Xi+1|2+λ3h∑i=0k-1|Xi|2+λ2h∑i=0k-1|Xi+1-m|2+λ4h∑i=0k-1|Xi-m|2+∑i=0k-1mi.
Note that
∑i=0k-1|Xi-m|2=∑i=-m-1|Xi|2+∑i=0k-m-1|Xi|2,∑i=0k-1|Xi+1-m|2=∑i=-m+1-1|Xi|2+∑i=0k-m|Xi|2.
Then, we have
|Yk|2≤|Y0|2+λ2h∑i=-m+1-1|Xi|2+λ4h∑i=-m-1|Xi|2-λ1h∑i=1k|Xi|2+λ3h∑i=0k-1|Xi|2+λ2h∑i=0k-m|Xi|2+λ4h∑i=0k-m-1|Xi|2+∑i=0k-1mi.
Namely,
|Yk|2+λ1h|Xk|2+λ2h∑i=k-m+1k-1|Xi|2+λ4h∑i=k-mk-1|Xi|2≤|Y0|2+λ1h|X0|2+λ2h∑i=-m+1-1|Xi|2+λ4h∑i=-m-1|Xi|2-(λ1-λ3-λ2-λ4)h∑i=0k-1|Xi|2+Mk,
where Mk=∑i=0k-1mi. By [18], Mk is a martingale with M0=0. From (3.3), we obtain that
λ1-λ3-λ2-λ4>0.
Using Lemma 2.2 yields
limk→∞∑i=0k-1|Xi|2<∞a.s.,h>0.
Then, we conclude that
limk→∞|Xk|2=0a.s.,h>0.
The proof is completed.

6. Numerical Experiments

In this section, we present numerical experiments to illustrate the theoretical results presented in the previous sections.

Consider the following scalar linear problemd[x(t)-12sin(x(t-2))]=(-8x(t)+sin(x(t-2)))dt+x(t-2)dB(t),t≥0,x(t)=t+1,-2≤t≤0.
For test (6.1), we have that L=72, λ1=11, λ2=4, λ3=0, and λ4=1. By Theorem 3.2, the exact solution to (6.1) is almost surely asymptotically stable.

Theorem 4.2 shows that the EM approximate solution to (6.1) can preserve the almost surely asymptotic stability of exact solutions for h<1/24. In Figure 1, we compute three different paths (Xn(ω1),Xn(ω2),Xn(ω3)) using EM (4.1) to approximate (6.1) with the stepsize h=1/25. Figure 1 shows that Xn(ω1),Xn(ω2),Xn(ω3) are asymptotically stable. Theorem 5.1 shows that the BEM approximate solution to (6.1) reproduces the almost surely asymptotic stability of exact solutions for any h>0. In Figure 2, three different paths (Xn(ω1),Xn(ω2),Xn(ω3)) are computed by using the BEM (5.1) to approximate (6.1) with the stepsize h=0.1. Figure 2 demonstrates that these paths are asymptotically stable.

Almost surely asymptotic stability of the EM approximate solution Xn with the stepsize h=1/25.

Almost surely asymptotic stability of the BEM approximate solution Xn with the stepsize h=0.1.

7. Conclusions

This paper deals with the almost surely asymptotic stability of Euler-type methods for NSDDEs by using the discrete semimartingale convergence theorem. We show that the EM reproduces the almost surely asymptotic stability of exact solutions to NSDDEs under an additional linear growth condition. If we assume the BEM is well defined, the BEM can also preserve the almost surely asymptotic stability without the additional linear growth condition.

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. This work is supported by the NSF of China (no.11071050).

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