This paper deals with nonlinear stochastic pantograph equations. For solving the equations, a class of extended Milstein methods are suggested. A mean-square stability criterion for this type of equations is presented. It is proved that under the suitable conditions the Milstein methods preserve the mean-square stability. Numerical examples further illustrate the obtained theoretical results.

1. Introduction

Stochastic delay differential equations (SDDEs) are often used to model some problems with aftereffect in many scientific fields such as physics, biology, mechanics, finance, and control theory. Generally speaking, it is hard to obtain the analytical solutions of SDDEs. Hence, recently, many researchers began to study their numerical solutions, and hence, some significant results have been achieved.

The stability analysis plays an important role in construction of excellent numerical algorithms for SDDEs. Hence, it has received wide attention of researchers. The early related results can be found in Mao [1, 2], Baker and Buckwar [3], Buckwar [4, 5], Küchler and Platen [6], and the references therein. More recently, for the linear SDDEs, Cao et al. [7], Liu et al. [8], and Wang and Zhang [9] studied mean-square stability (MS-stability) of Euler-Maruyama, semi-implicit Euler-Maruyama, and Milstein methods, respectively. Taking use of the Halanay inequality, Baker and Buckwar [10] extended the MS-stability analysis of Euler-Maruyama methods to nonlinear SDDEs. Moreover, Wang and Zhang [11] also dealt with nonlinear MS-stability of Milstein methods.

We note that the above numerical stability investigations were mainly devoted to the case of constant delay. Although the deterministic delay differential equations with variable delays have been widely studied (see, e.g., [12, 13] and the references therein), the case of variable delay of SDDEs was rarely concerned. Fan and Liu [14] first studied linear stochastic pantograph equations and gave MS-stability criteria of semi-implicit Euler methods. Also, by taking use of the analytical and discrete Razumikhin theorems, they dealt with -moment stability of linear stochastic pantograph equations and their semi-implicit Euler method (cf. [15]). Recently, Xiao et al. [16, 17] gave sufficient MS-stability conditions of backward Euler method and semi-implicit Euler method with variable stepsize for linear stochastic pantograph differential equations. In the present paper, we will investigate the MS-stability of nonlinear stochastic pantograph equations and their Milstein methods. Some criteria for MS-stability of the analytical and numerical solutions will be derived. Numerical experiments will be used to illustrate the obtained theoretical results.

2. MS-Stability of the Analytical Solutions

Let be a complete probability space with a filtration , which is right-continuous and satisfies that each contains all -null sets in , and is a one-dimensional Brownian motion defined on the probability space. Moreover, we introduce the following notations:

Consider the following nonlinear stochastic pantograph equations: where is a -value random process, denotes a given constant, and are two given Borel-measurable functions, is an -measurable -value random variable, and . Throughout this paper, we always assume that (2) has a unique solution .

Definition 1. The solution of (2) is said to be MS-stable if

Theorem 2. Assume that there exist constants , , and such that Then, the solution of (2) is MS-stable whenever

Proof. By the Itô formula (cf. [1]), we have Integrating from 0 to on both sides of the equality (7) and then taking expectation yield that Since , we further have Applying the conditions (4) and (5) to (9), it follows that which gives This, together with (6), implies . Therefore, the theorem is proven.

3. MS-Stability of the Numerical Solutions

For the stability analysis, we introduce the following notational conventions: where , denoting independent -distributed Gaussian random variables. Moreover, on space , we define an inner product and the corresponding induced norm as follows: where , .

Applying the Milstein method to (2) derives the following numerical scheme: where is the computational stepsize, , and , , and are approximations to , , and , respectively. When set where and , the approximations of and can be defined as follows: In this way, an extended Milstein method, composed by (14) and (16), is obtained.

Definition 3. An extended Milstein method (14)–(16) is said to be MS-stable if there exists an such that

Lemma 4. The Itô-type double integrals , have the following properties:

Proof. The equalities can be derived directly from the properties of martingales. Moreover, by the equality , we have Also, it follows from the properties of Itô integral that This completes the proof.

Let . Then, we have the following lemma.

Lemma 5. Assume that there exist positive integers , and such that Then, the sequence is monoincreasing and has at most equal components.

Proof. It follows from that Let denote the integer part of a real number. Then, by and , we have for all that Hence, it holds that This shows that the sequence is monoincreasing.
For proving the second part of this lemma, we use reduction to absurdity. If the sequence has components which satisfy that then, by , , and , we have This is contrary to . Hence, Lemma 5 is proven.

With the above lemmas, the main result can be stated as follows.

Theorem 6. Assume that the conditions (4) and (5) hold and that there exist constants , , and such that Then, the extended Milstein method (14)–(16) is MS-stable whenever where

Proof. By (14), we have Using conditions (4) and (27) generates respectively. Moreover, the -measurability implies that and a combination of Lemma 4, (5), and (28) gives Taking expectation on both sides of (31) and then substituting (32)–(36) into the obtained inequality yield Combining (16) and (37) derives An induction to (38) yields Applying Lemma 5 to (39), it follows that This shows that the positive series is bounded when (29) holds and , where Therefore, it holds that . This completes the proof.

4. Numerical Illustration

In this section, we give a numerical example to illustrate the obtained theoretical results. Consider the following stochastic pantograph equation: It is easy to verify that the conditions of Theorems 2 and 6 can be satisfied with parameters Hence, both the solution of (42) and its solving method (14) are all MS-stable.

Applying the extended Milstein method (14)–(16), with stepsizes , respectively, to (42) on interval , we can obtain four groups of numerical solutions (see Figure 1), where we take the average of 1000 block samples. Figure 1 shows that the numerical solutions are all stable. However, if we take a larger stepsize, then the numerical stability cannot be assured. This is shown in Figure 2, where stepsizes are used, which leads to two groups of unstable solutions.

5. Conclusions

In this paper, a class of extended Milstein methods for solving nonlinear stochastic pantograph equations are suggested. A mean-square stability criterion for this type of equations is presented. It is proved that, under the suitable conditions, if the stepsize satisfies the sufficient condition , where is given by (41), then the Milstein methods preserve the mean-square stability. How does one obtain an exact critical stepsize such that the method is stable for and unstable for ? This is a difficult problem which keeps open at present. We will work on it in the future research.


This work is supported by NSFC (nos. 11171125, 91130003, 11201162, and 11301099), NSFH (no. 2011CDB289), Humanities and Social Science Research Projects in Ministry of Education (13YJA910003), the Doctor Scientific Research Foundation of Guangxi Normal University, and the Fundamental Research Funds for the Central Universities, HUST (2011QN168).