Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 724241, 7 pages

http://dx.doi.org/10.1155/2013/724241

## Mean-Square Stability of Milstein Methods for Stochastic Pantograph Equations

^{1}School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China^{2}School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China

Received 1 June 2013; Revised 22 September 2013; Accepted 27 September 2013

Academic Editor: Guo-Cheng Wu

Copyright © 2013 Feiyan Xiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### Acknowledgments

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).

#### References

- X. Mao,
*Stochastic Differential Equations and Their Applications*, Horwood Publishing Limited, Chichester, UK, 1997. View at MathSciNet - X. Mao and S. Sabanis, “Numerical solutions of stochastic differential delay equations under local Lipschitz condition,”
*Journal of Computational and Applied Mathematics*, vol. 151, no. 1, pp. 215–227, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. T. H. Baker and E. Buckwar, “Numerical analysis of explicit one-step methods for stochastic delay differential equations,”
*LMS Journal of Computation and Mathematics*, vol. 3, pp. 315–335, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Buckwar, “Introduction to the numerical analysis of stochastic delay differential equations,”
*Journal of Computational and Applied Mathematics*, vol. 125, no. 1-2, pp. 297–307, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Buckwar, “One-step approximations for stochastic functional differential equations,”
*Applied Numerical Mathematics*, vol. 56, no. 5, pp. 667–681, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - U. Küchler and E. Platen, “Strong discrete time approximation of stochastic differential equations with time delay,”
*Mathematics and Computers in Simulation*, vol. 54, no. 1–3, pp. 189–205, 2000. View at Publisher · View at Google Scholar · View at MathSciNet - W. Cao, M. Liu, and Z. Fan, “MS-stability of the Euler-Maruyama method for stochastic differential delay equations,”
*Applied Mathematics and Computation*, vol. 159, no. 1, pp. 127–135, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Liu, W. Cao, and Z. Fan, “Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation,”
*Journal of Computational and Applied Mathematics*, vol. 170, no. 2, pp. 255–268, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wang and C. Zhang, “An analysis of stability of Milstein method for stochastic differential equations with delay,”
*Computers & Mathematics with Applications*, vol. 51, no. 9-10, pp. 1445–1452, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. T. H. Baker and E. Buckwar, “Exponential stability in $p$-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations,”
*Journal of Computational and Applied Mathematics*, vol. 184, no. 2, pp. 404–427, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Y. Wang and C. J. Zhang, “Mean-square stability of the Milstein method for solving nonlinear stochastic delay differential equations,”
*Mathematica Applicata*, vol. 21, no. 1, pp. 201–206, 2008. View at Google Scholar · View at MathSciNet - C. Zhang and G. Sun, “The discrete dynamics of nonlinear infinite-delay-differential equations,”
*Applied Mathematics Letters*, vol. 15, no. 5, pp. 521–526, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Zhang and G. Sun, “Nonlinear stability of Runge-Kutta methods applied to infinite-delay-differential equations,”
*Mathematical and Computer Modelling*, vol. 39, no. 4-5, pp. 495–503, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-C. Fan and M.-Z. Liu, “The asymptotically mean square stability of the linear stochastic pantograph equation,”
*Mathematica Applicata*, vol. 20, no. 3, pp. 519–523, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Fan, M. Song, and M. Liu, “The $\alpha $th moment stability for the stochastic pantograph equation,”
*Journal of Computational and Applied Mathematics*, vol. 233, no. 2, pp. 109–120, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Xiao and H. Zhang, “Convergence and stability of numerical methods with variable step size for stochastic pantograph differential equations,”
*International Journal of Computer Mathematics*, vol. 88, no. 14, pp. 2955–2968, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Xiao, M. Song, and M. Liu, “Convergence and stability of the semi-implicit Euler method with variable stepsize for a linear stochastic pantograph differential equation,”
*International Journal of Numerical Analysis and Modeling*, vol. 8, no. 2, pp. 214–225, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet