#### Abstract

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 Taking the environmental disturbances into account, we are led to the following neutral stochastic pantograph equation (NSPE) 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 be a complete probability space with filtration satisfying the usual conditions (i.e., it is right continuous, and contains all -null sets). is a scalar Brownian motion defined on the probability space. denotes the Euclidean norm in . The inner product of in is denoted by or . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by . Let denote the family of all -value measurable -adapted processes such that w.p.1. Let denote the family of all -value measurable -adapted processes such that w.p.1.

Consider an -dimensional neutral stochastic pantograph equation on with -measurable bounded initial data . Here , , , and .

Let denote the family of continuous functions from to . Let denote the family of all nonnegative functions on which are continuously once differentiable in and twice differentiable in . For each , define an operator from to by where

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

*Definition 2.1. *A -value stochastic process on is called a solution of (2.1) if it has the following properties:(1) is continuous and -adapted;(2), :(3), and (2.1) holds for every with probability 1.A solution is said to be unique if any other solution is indistinguishable from it, that is,

To ensure the existence and uniqueness of the solution to (2.1) on , we impose the following assumptions on the coefficients , and .

*Assumption 2.2. *Assume that both and satisfy the global Lipschitz condition and the linear growth condition. That is, there exist two positive constants and such that for all , and ,
and for all , and ,

*Assumption 2.3. *Assume that there is a constant such that

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 be a solution to (2.1) with -measurable bounded initial data . Then
*

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 -measurable bounded initial data , (2.1) has a unique solution on .*

Based on Lemma in [21] and Lemma 2.4, this theorem can be proved in the same way as Theorem 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 and be two sequences of nonnegative random variables such that both and are -measurable for , and a.s. Let be a real-valued local martingale with a.s. Let be a nonnegative -measurable random variable. Assume that is a nonnegative semimartingale with the Doob-Mayer decomposition
** a.s., then for almost all :
**
that is, both and 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 -measurable bounded initial data . Moreover, we always assume that in the following sections. Therefore, (2.1) admits a trivial solution .

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

*Definition 3.1. *The solution to (2.1) is said to be almost surely asymptotically stable if
for any bounded -measurable bounded initial data .

Lemma 3.2. *Let and be a continuous functions. Assume that
**
Then,
*

*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 for given -measurable bounded initial data . Let Assumption 2.3 hold. Assume that there are functions , , and four positive constants such that
**
Then, for any **
where is positive and satisfies
**
That is,
*

*Proof. *Choose for and . 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 for given -measurable bounded initial data . Let Assumption 2.3 hold. Assume that there are four positive constants such that
**
for and . If
**
then, the global solution to (2.1) is almost surely asymptotically stable.*

*Proof. *Let . Applying Lemma 3.3 and Lemma 3.2 with , 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 as follows. Let , . Set and . We define grid points in by where and define the other grid points by It is easy to see that the grid point satisfies for , and the step size satisfies For the given mesh , we define the BEM for (2.1) as follows: Here, is an approximation value of and -measurable. is the Brownian increment. The approximations are calculated by the following formulae: where . 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 to the BEM (4.4) is said to be almost surely asymptotically stable if
for any bounded -measurable bounded initial data .

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
**
That is, the approximate solution to the BEM (4.4) is almost surely asymptotically stable.*

*Proof. *Set . For , from (4.4), we have
Then, we can obtain that
which subsequently leads to
where
By conditions (3.8) and (3.9), we have
Using the equality , we obtain that
Inserting these inequalities to (4.12) and using Assumption 2.3 yield
Let , , , and . Using these notations, (4.14) implies that
Then, we can conclude that
Note that
We, therefore, have
Similar to (4.15), from (4.4), we can obtain that
where are defined as before,
From (4.19), we have
where
Obviously . By (4.18) and (4.21), we can obtain that
where . Similar to the proof in [18], we can obtain that is a martingale with . Note that and for . Then, we have
Using the condition (3.9) and , we obtain that there exists an integer such that
Set ,
Obviously,
Moreover, (4.23) implies that
Here . According to (4.27), using Lemma 2.6 yields
Then, we have
Note that
We therefore obtain that
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: For the test (5.1), we have , , , and 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 () using the BEM (4.4) with . In Figure 2, three different paths () of BEM approximations are computed with . The results demonstrate that these paths are asymptotically stable.

#### Acknowledgments

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