We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph equations (NSPEs), and sufficient conditions are obtained. Based on these sufficient conditions, we show that the backward Euler method (BEM) with variable stepsize can preserve the almost surely asymptotic stability. Numerical examples are demonstrated for illustration.

1. Introduction

The neutral pantograph equation (NPE) plays important roles in mathematical and industrial problems (see [1]). It has been studied by many authors numerically and analytically. We refer to [1–7]. One kind of NPEs reads[x(t)-N(x(qt))]′=f(t,x(t),x(qt)).
Taking the environmental disturbances into account, we are led to the following neutral stochastic pantograph equation (NSPE)d[x(t)-N(x(qt))]=f(t,x(t),x(qt))dt+g(t,x(t),x(qt))dB(t),
which is a kind of neutral stochastic delay differential equations (NSDDEs).

Using the continuous semimartingale convergence theorem (cf. [8]), Mao et al. (see [9, 10]) studied the almost surely asymptotic stability of several kinds of NSDDEs. As most NSDDEs cannot be solved explicitly, numerical methods have become essential. Efficient numerical methods for NSDDEs can be found in [11–13]. The stability theory of numerical solutions is one of fundamental research topics in the numerical analysis. The almost surely asymptotic stability of numerical solutions for stochastic differential equations (SDEs) and stochastic functional differential equations (SFDEs) has received much more attention (see [14–19]). Corresponding to the continuous semimartingale convergence theorem (cf. [8]), the discrete semimartingale convergence theorem (cf. [17, 20]) also plays important roles in the almost surely asymptotic stability analysis of numerical solutions for SDEs and SFDEs (see [17–19]). To our best knowledge, no results on the almost surely asymptotic stability of exact and numerical solutions for the NSPE (1.2) can be found. We aim in this paper to study the almost surely asymptotic stability of exact and numerical solutions to NSPEs by using the continuous semimartingale convergence theorem and the discrete semimartingale convergence theorem. We prove that the backward Euler method (BEM) with variable stepsize can preserve the almost surely asymptotic stability under the conditions which guarantee the almost surely asymptotic stability of the exact solution.

In Section 2, we introduce some necessary notations and elementary theories of NSPEs (1.2). Moreover, we state the discrete semimartingale convergence theorem as a lemma. In Section 3, we study the almost surely asymptotic stability of exact solutions to NSPEs (1.2). Section 4 gives the almost surely asymptotic stability of the backward Euler method with variable stepsize. Numerical experiments are presented in the finial section.

2. Neutral Stochastic Pantograph Equation

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 ℒ1([0,T];Rn) denote the family of all Rn-value measurable ℱt-adapted processes f={f(t)}0≤t≤T such that ∫0T|f(t)|dt<∞ w.p.1. Let ℒ2([0,T];Rn) denote the family of all Rn-value measurable ℱt-adapted processes f={f(t)}0≤t≤T such that ∫0T|f(t)|2dt<∞ w.p.1.

Consider an n-dimensional neutral stochastic pantograph equationd[x(t)-N(x(qt))]=f(t,x(t),x(qt))dt+g(t,x(t),x(qt))dB(t),
on t≥0 with ℱ0-measurable bounded initial data x(0)=x0. Here 0<q<1, f:R+×Rn×Rn→Rn, g:R+×Rn×Rn→Rn, and N:Rn→Rn.

Let C(Rn;R+) denote the family of continuous functions from Rn to R+. 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.

To be precise, we first give the definition of the solution to (2.1) on 0≤t≤T.

Definition 2.1.

A Rn-value stochastic process x(t) on 0≤t≤T is called a solution of (2.1) if it has the following properties:

x(0)=x0, and (2.1) holds for every t∈[0,T] with probability 1.

A solution x(t) is said to be unique if any other solution x¯(t) is indistinguishable from it, that is,
P{x(t)=x¯(t),0≤t≤T}=1.

To ensure the existence and uniqueness of the solution to (2.1) on t∈[0,T], we impose the following assumptions on the coefficients N,f, and g.

Assumption 2.2.

Assume that both f and g satisfy the global Lipschitz condition and the linear growth condition. That is, there exist two positive constants L and K such that for all x,y,x¯,y¯∈Rn, and t∈[0,T],
|f(t,x,y)-f(t,x¯,y¯)|2∨|g(t,x,y)-g(t,x¯,y¯)|2≤L(|x-x¯|2+|y-y¯|2),
and for all x,y∈Rn, and t∈[0,T],
|f(t,x,y)|2∨|g(t,x,y)|2≤K(1+|x|2+|y|2).

Assumption 2.3.

Assume that there is a constant κ∈(0,1) such that
|N(x)-N(y)|≤κ|x-y|,∀x,y∈Rn.

Under Assumptions 2.2 and 2.3, the following results can be derived.

Lemma 2.4.

Let Assumptions 2.2 and 2.3 hold. Let x(t) be a solution to (2.1) with ℱ0-measurable bounded initial data x(0)=x0. Then
E(sup0≤t≤T|x(t)|2)≤(1+(1-κ)κ+3(1-κ)(1-κ)2(1-κ)E|x0|2)exp{6K(T+4)T(1-κ)(1-κ)}.

The proof of Lemma 2.4 is similar to Lemma 6.2.4 in [21], so we omit the details.

Theorem 2.5.

Let Assumptions 2.2 and 2.3 hold, then for any ℱ0-measurable bounded initial data x(0)=x0, (2.1) has a unique solution x(t) on t∈[0,T].

Based on Lemma 6.2.3 in [21] and Lemma 2.4, this theorem can be proved in the same way as Theorem 6.2.2 in [21], so the details are omitted.

The discrete semimartingale convergence theorem (cf. [17, 20]) will play an important role in this paper.

Lemma 2.6.

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.Iflimi→∞Ai<∞ a.s., then for almost all ω∈Ω:
limi→∞Xi<∞,limi→∞Ui<∞,
that is, both Xi and Ui converge to finite random variables.

3. Almost Surely Asymptotic Stability of Neutral Stochastic Pantograph Equations

In this section, we investigate the almost surely asymptotic stability of (2.1). We assume (2.1) has a continuous unique global solution for given ℱ0-measurable bounded initial data x0. Moreover, we always assume that f(t,0,0)=0,g(t,0,0)=0,N(0)=0 in the following sections. Therefore, (2.1) admits a trivial solution x(t)=0.

To be precise, let us give the definition on the almost surely asymptotic stability of (2.1).

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 bounded ℱ0-measurable bounded initial data x(0).

Lemma 3.2.

Let ρ:R+→(0,∞) and z:[0,∞)→Rn be a continuous functions. Assume that
σ1∶=limsupt→∞ρ(t)ρ(qt)<1κ,σ2∶=limsupt→∞[ρ(t)|z(t)-N(z(qt))|]<∞.
Then,
limsupt→∞[ρ(t)|z(t)|]≤σ21-κσ1.

Proof.

Using the idea of Lemma 3.1 in [9], we can obtain the desired result.

Lemma 3.3.

Suppose that (2.1) has a continuous unique global solution x(t) for given ℱ0-measurable bounded initial data x0. Let Assumption 2.3 hold. Assume that there are functions U∈C1,2(R+×Rn;R+), w∈C(Rn;R+), and four positive constants λ1>λ2,λ3,λ4 such that
LU(t,x,y)≤-λ1w(x)+qλ2w(y),(t,x,y)∈R+×Rn×Rn,U(t,x-N(y))≤λ3w(x)+λ4w(y),(t,x)∈R+×Rn.
Then, for any ɛ∈(0,γ*)limsupt→∞t(γ*-ɛ)U(t,x(t)-N(x(qt)))<∞a.s.,
where γ* is positive and satisfies
λ1=λ2q-γ*.
That is,
limt→∞U(t,x(t)-N(x(qt)))=0a.s.

Proof.

Choose V(t,x(t))=tγU(t,x(t)-N(x(qt))) for (t,x)∈R+×Rn and γ>0. Similar to the proof of Lemma 2.2 in [9], the desired conclusion can be obtained by using the continuous semimartingale convergence theorem (cf. [8]).

Theorem 3.4.

Suppose that (2.1) has a continuous unique global solution x(t) for given ℱ0-measurable bounded initial data x0. Let Assumption 2.3 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+λ4q,
then, the global solution x(t) to (2.1) is almost surely asymptotically stable.

Proof.

Let U(t,x)=w(x)=|x|2. Applying Lemma 3.3 and Lemma 3.2 with ρ=1, we can obtain the desired conclusion.

Theorem 3.4 gives sufficient conditions of the almost surely asymptotic stability of NSPEs (2.1). Based on this result, we will investigate the almost surely asymptotic stability of the BEM with variable stepsize for (2.1) in the following section.

4. Almost Surely Asymptotic Stability of the Backward Euler Method

To define the BEM for (2.1), we introduce a mesh H={m;t-m,t-m+1,…,t0,t1,…,tn,…} as follows. Let hn=tn+1-tn, h-m-1=t-m. Set t0=γ0>0 and tm=q-1γ0. We define m-1 grid points t1<t2<⋯<tm-1 in (t0,tm) by ti=t0+iΔ0,fori=1,2,…,m-1,
where Δ0=(tm-t0)/m and define the other grid points by tkm+i=q-kti,fork=-1,0,1,…,i=0,1,2,…,m-1.
It is easy to see that the grid point tn satisfies qtn=tn-m for n≥0, and the step size hn satisfies qhn=hn-m,forn≥0,limn→∞hn=∞.
For the given mesh H, we define the BEM for (2.1) as follows:Yn+1-N(Yn+1-m)=Yn-N(Yn-m)+hnf(tn+1,Yn+1,Yn+1-m)+g(tn,Yn,Yn-m)ΔBn,n≥-m,Y-m-N(Y-m-m)=x0-N(x0)+h-m-1f(t-m,Y-m,Y-m-m)+g(0,x0,x0)B(t-m).
Here, Yn(n≥-m) is an approximation value of x(tn) and ℱtn-measurable. ΔBn=B(tn+1)-B(tn) is the Brownian increment. The approximations Yn-m(n=-m,-m+1,…,-1) are calculated by the following formulae:
Yn-m=(1-θn)x0+θnY-m,n=-m,-m+1,…,-1,
where θn=qtn/t-m. As a standard hypothesis, we assume that the BEM (4.4) is well defined.

To be precise, let us introduce the definition on the almost surely asymptotic stability of the BEM (4.4).

Definition 4.1.

The approximate solution Yn to the BEM (4.4) is said to be almost surely asymptotically stable if
limn→∞Yn=0a.s.
for any bounded ℱ0-measurable bounded initial data x0.

Theorem 4.2.

Assume that the BEM (4.4) is well defined. Let Assumption 2.3 hold. Let conditions (3.8) and (3.9) hold. Then the BEM approximate solution (4.4) obeys
limn→∞Yn=0a.s.
That is, the approximate solution Yn to the BEM (4.4) is almost surely asymptotically stable.

Proof.

Set Y¯n=Yn-N(Yn-m). For n≥0, from (4.4), we have
|Y¯n+1-hnf(tn+1,Yn+1,Yn+1-m)|2=|Y¯n+g(tn,Yn,Yn-m)ΔBn|2.
Then, we can obtain that
|Y¯n+1|2≤|Y¯n|2+2hn〈Y¯n+1,f(tn+1,Yn+1,Yn+1-m)〉+|g(tn,Yn,Yn-m)ΔBn|2+2〈Y¯n,g(tn,Yn,Yn-m)〉ΔBn,
which subsequently leads to
|Y¯n+1|2≤|Y¯n|2+2hn〈Y¯n+1,f(tn+1,Yn+1,Yn+1-m)〉+|g(tn,Yn,Yn-m)|2hn+m¯n,
where
m¯n=2〈Y¯n,g(tn,Yn,Yn-m)〉ΔBn+|g(tn,Yn,Yn-m)|2(ΔBn2-hn).
By conditions (3.8) and (3.9), we have
|Y¯n+1|2≤|Y¯n|2-λ1hn|Yn+1|2+λ2hn|Yn+1-m|2+(λ3|Yn|2+λ4|Yn-m|2)hn+m¯n.
Using the equality |a+b|2≤2|a|2+2|b|2, we obtain that
|Y¯n+1|2≥12|Yn+1|2-|N(Yn+1-m)|2,|Y¯n|2≤2|Yn|2+2|N(Yn-m)|2.
Inserting these inequalities to (4.12) and using Assumption 2.3 yield
(12+λ1hn)|Yn+1|2≤(2+λ3hn)|Yn|2+(κ2+λ2hn)|Yn+1-m|2+(2κ2+λ4hn)|Yn-m|2+m¯n.
Let An=1+2λ1hn, Bn=3-2λ1hn+2λ3hn, Cn=2κ2+2λ2hn, and Dn=4κ2+2λ4hn. Using these notations, (4.14) implies that
|Yn+1|2-|Yn|2≤BnAn|Yn|2+CnAn|Yn+1-m|2+DnAn|Yn-m|2+2Anm¯n.
Then, we can conclude that
|Yn|2≤|Y0|2+∑i=0n-1BiAi|Yi|2+∑i=0n-1CiAi|Yi+1-m|2+∑i=0n-1DiAi|Yi-m|2+∑i=0n-12Aim¯i.
Note that
∑i=0n-1CiAi|Yi+1-m|2=∑i=-m+1n-mCi+m-1Ai+m-1|Yi|2=∑i=-m+1-1Ci+m-1Ai+m-1|Yi|2+∑i=0n-1Ci+m-1Ai+m-1|Yi|2-∑i=n-m+1n-1Ci+m-1Ai+m-1|Yi|2,∑i=0n-1DiAi|Yi-m|2=∑i=-mn-m-1Di+mAi+m|Yi|2=∑i=-m-1Di+mAi+m|Yi|2+∑i=0n-1Di+mAi+m|Yi|2-∑i=n-mn-1Di+mAi+m|Yi|2.
We, therefore, have
|Yn|2+∑i=n-m+1n-1Ci+m-1Ai+m-1|Yi|2+∑i=n-mn-1Di+mAi+m|Yi|2≤|Y0|2+∑i=-m+1-1Ci+m-1Ai+m-1|Yi|2+∑i=-m-1Di+mAi+m|Yi|2+∑i=0n-1(BiAi+Ci+m-1Ai+m-1+Di+mAi+m)|Yi|2+∑i=0n-12Aim¯i.
Similar to (4.15), from (4.4), we can obtain that
|Y0|2-|Y-1|2≤B-1A-1|Y-1|2+C-1A-1|Y-m|2+D-1A-1[2(1-θ-1)2|x0|2+2θ-12|Y-m|2]+2A-1m¯-1,|Yn|2-|Yn-1|2≤Bn-1An-1|Yn-1|2+Cn-1An-1[2(1-θn)2|x0|2+2θn2|Y-m|2]+Dn-1An-1[2(1-θn-1)2|x0|2+2θn-12|Y-m|2]+2An-1m¯n-1,-m+1≤n≤-1,|Y-m|2-|x0|2≤B-m-1+D-m-1A-m-1|x0|2+C-m-1A-m-1[2(1-θ-m)2|x0|2+2θ-m2|Y-m|2]+2A-m-1m¯-m-1,
where An,Bn,Cn,Dn(n=-m-1,…,-1) are defined as before,
m¯n=2〈Yn-N(Yn-m),g(tn,Yn,Yn-m)〉ΔBn+|g(tn,Yn,Yn-m)|2(ΔBn2-hn),-m≤n≤-1,m¯-m-1=2〈x0-N(x0),g(0,x0,x0)〉B(t-m)+|g(0,x0,x0)|2(B2(t-m)-t-m).
From (4.19), we have
|Y0|2≤A|x0|2+B|Y-m|2+∑i=-m+1-1BiAi|Yi|2+∑i=-m-1-12Aim¯i,
where
A=1+B-m-1+D-m-1A-m-1+∑i=-m-12DiAi((1-θi)2)+∑i=-m-1-22CiAi((1-θi+1)2),B=B-mA-m+B-1A-1+∑i=-m-12DiAiθi2+∑i=-m-1-22CiAiθi+12.
Obviously A>0. By (4.18) and (4.21), we can obtain that
|Yn|2+∑i=n-m+1n-1Ci+m-1Ai+m-1|Yi|2+∑i=n-mn-1Di+mAi+m|Yi|2≤A|x0|2+(B+D0A0)|Y-m|2+∑i=-m+1n-1(BiAi+Ci+m-1Ai+m-1+Di+mAi+m)|Yi|2+Mn,
where Mn=∑i=-m-1n-1(2/Ai)m¯i. Similar to the proof in [18], we can obtain that Mn is a martingale with M-m-1=0. Note that hi+m-1≤hi+m and hi+m=hi/q for i≥-m. Then, we have
BiAi+Ci+m-1Ai+m-1+Di+mAi+m≤(3+6κ2)-2(λ1hi-λ3hi-λ2hi+m-1-λ4hi+m)1+2λ1hi≤11+2λ1hi{(3+6κ2)-2(λ1-λ3-λ2q-λ4q)hi}.
Using the condition (3.9) and limi→∞hi=∞, we obtain that there exists an integer i* such that
BiAi+Ci+m-1Ai+m-1+Di+mAi+m≥0,i≤i*,BiAi+Ci+m-1Ai+m-1+Di+mAi+m<0,i>i*.
Set U-m-1=0,
U-m={(B+D0A0)|Y-m|2if(B+D0A0)>0,0if(B+D0A0)≤0,Un={U-m+∑i=-m+1n-1(BiAi+Ci+m-1Ai+m-1+Di+mAi+m)|Yi|2if-m+1≤n≤i*+1,U-m+∑i=-m+1i*(BiAi+Ci+m-1Ai+m-1+Di+mAi+m)|Yi|2ifn>i*+1,Vn={0if-m-1≤n≤i*+1,-∑i=i*+1n-1(BiAi+Ci+m-1Ai+m-1+Di+mAi+m)|Yi|2ifn>i*+1.
Obviously,
limn→∞Un=U-m+∑i=-m+1i*(BiAi+Ci+m-1Ai+m-1+Di+mAi+m)|Yi|2<∞,a.s.
Moreover, (4.23) implies that
|Yn|2≤C|x0|2+Un-Vn+Mn.
Here C=max{A,1}. According to (4.27), using Lemma 2.6 yields
limsupn→∞|Yn|2<∞a.s.,limn→∞Vn<∞a.s.
Then, we have
limi→∞-(BiAi+Ci+m-1Ai+m-1+Di+mAi+m)|Yi|2=0a.s.
Note that
limi→∞-(BiAi+Ci+m-1Ai+m-1+Di+mAi+m)=λ1-λ2-λ3-λ4λ1>0.
We therefore obtain that
limn→∞|Yn|2=0a.s.
Then, the desired conclusion is obtained. This completes the proof.

5. Numerical Experiments

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

Consider the following scalar problem:d[x(t)-12x(0.5t)]=(-8x(t)+x(0.5t))dt+sin(x(0.5t))dB(t),t≥0,x(0)=x0.
For the test (5.1), we have λ1=11, λ2=4, λ3=0, and λ4=1 corresponding to Theorem 3.4. By Theorem 3.4, the solution to (5.1) is almost surely asymptotically stable.

Theorem 4.2 shows that the BEM approximation to (5.1) is almost surely asymptotically stable. In Figure 1, We compute three different paths (Yn(ω1),Yn(ω2),Yn(ω3)) using the BEM (4.4) with x0=2,t0=0.01,m=2. In Figure 2, three different paths (Yn(ω1),Yn(ω2),Yn(ω3)) of BEM approximations are computed with x0=10,t0=1,m=1. The results demonstrate that these paths are asymptotically stable.

Almost surely asymptotic stability with x0=2,t0=0.01,m=2.

Almost surely asymptotic stability with x0=10,t0=1,m=1.

Acknowledgments

The author would like to thank the referees for their helpful comments and suggestions.

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