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Journal of Applied Mathematics

Volume 2014, Article ID 545830, 10 pages

http://dx.doi.org/10.1155/2014/545830
Research Article

T-Stability of the Heun Method and Balanced Method for Solving Stochastic Differential Delay Equations

School of Mathematics, Hefei University of Technology, Hefei 230009, China

Received 9 March 2014; Accepted 15 May 2014; Published 11 June 2014

Academic Editor: Senlin Guo

Copyright © 2014 Xiaolin Zhu and Hu Peng. 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 studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs). Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.

1. Introduction

Stochastic differential delay equations (SDDEs) are the promotion of stochastic differential equations (SDEs) and differential delay equations (DDEs). These kinds of equations consider not only the stochastic factors in the process of the development of a system, but also the impact of the delay. As an important mathematical model, SDDEs have been applied widely in many areas, such as stochastic control, economics, and biology. Since it is difficult to find the analytic solutions to SDDEs, to get the numerical solutions to SDDEs generated by some numerical methods is commonly used. For a numerical method, it is important to analyze its stability.

The general form of SDDEs with Gaussian white noise is where , , , and is a standard -dimensional Wiener process. Equation (1) has a unique solution if and are sufficiently smooth and satisfy the following conditions: where , and and are constants. The condition in (2) is called Lipschitz condition, and the condition in (3) is called the linear growth condition.

The main numerical methods for SDDEs are Euler-Maruyama method [1, 2] and Milstein method [3] at present. The mean square stability of these methods for SDDEs has been well studied. Cao et al. [1] studied the mean square stability of Euler-Maruyama method for linear SDDEs. Liu et al. [2] studied the mean square stability of the semi-implicit Euler method for linear SDDEs. Wang and Zhang [3] discussed the mean square stability of Milstein method for linear SDDEs. Wang and Chen [4, 5] studied the mean square stability of semi-implicit Euler method for nonlinear neutral SDDEs and that of Heun methods for nonlinear SDDEs. Tan et al. [6] discussed the mean square stability of balanced methods for SDDEs. T-stability is introduced by Saito et al. in [79], and it is another kind of stability with respect to the approximate sequence of sample path. Cao [10] studied the T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise. Rathinasamy and Balachandran [11] studied the T-stability of the split-step θ-methods for linear SDDEs. Yang and Liu [12] discussed the T-stability of the θ-method for a stochastic pantograph differential equation.

Applying the Heun method [5] to (1) gives where is a step size with for a positive integer and . is an approximation of , and if . .

The mean square stability of the Heun method in (4) was studied in [5], but there is no result about T-stability of the method at present. This paper gives two T-stable conditions of the Heun method (4) for two kinds of linear SDDEs.

The balanced method for solving (1) is where and are real matrix functions.

Let , where is a unit matrix, , , , and . Assume that is invertible with .

The mean square stability of the balanced method for SDEs and SDEs with jumps was studied in [13, 14], respectively. In 2011, Tan et al. [6] applied the balanced method to SDDEs and discussed the mean square convergence and stability of this method. However, there is no research result about T-stability of the balanced method (5) at present. In this paper, the conditions under which the balanced method (5) is T-stable are obtained for two kinds of linear SDDEs.

Section 2 introduces the stochastically asymptotically stable conditions in the large for two kinds of linear SDDEs. In Section 3, T-stability of the Heun method equipped with a specified driving process is discussed and the corresponding step size range is given. Section 4 studies T-stability of the balanced method, and Section 5 uses some numerical examples to verify the results given in this paper.

2. Asymptotic Stability of Analytical Solution

Consider the following two scalar linear test equations: Let be a complete probability space with a filtration , which is right continuous, and each contains all -null sets in   . In (6) and (7), ; is a one-dimensional standard Wiener process; initial function is -measurable and . Equations (6) and (7) have unique strong solution if (6) and (7) meet Lipschitz condition in (2) and the linear growth condition in (3).

Definition 1 (see [10]). The solution of (1) is stochastically asymptotically stable in the large if for all initial functions .

From Corollary in [15], we get Lemmas 2 and 3 as follows.

Lemma 2. The solution of (6) is stochastically asymptotically stable in the large if parameters , and in (6) satisfy

Lemma 3. The solution of (7) is stochastically asymptotically stable in the large if parameters , and in (7) satisfy

3. T-Stability of the Heun Method

Definition 4 (see [10]). Suppose that the condition in (9) or in (10) is fulfilled. A numerical scheme equipped with a specified driving process is said to be T-stable if for the driving process, where is the numerical solution generated by the numerical scheme applied to the test equation (6) or (7).

For analyzing T-stability, we focus our attention on the trajectory of numerical solution. A specified driving process proposed in Definition 4 is used to approximate the Wiener increment of the numerical methods. This paper analyzes the Heun method equipped with two-point random variables for the driving process. So and , where denotes the probability.

The Heun method applied to (6) and (7), respectively, gives where is a step size with for a positive integer and , if , .

For the Heun method in (12), we have Let It is clear that if , and therefore the Heun method in (12) is T-stable. Denote Since ’s follow two-point distribution, we get if , which means that the Heun method in (12) is T-stable.

Similarly, for the Heun method in (13), we can get as follows: and if , which means that the Heun method in (13) is T-stable.

Theorem 5. Suppose (6) meets the condition in (9). The Heun method in (12) is T-stable if , where

Proof. The condition in (9) gives . Denote and .

For , we have , from the condition in (9). Let

Only the conclusion of the theorem when is proven, and that of the theorem when or or can be proven similarly.

If , then and , and we have and . Hence Since and , we have and Consequently, which means that the Heun method (12) is T-stable if .

(a) When , if , then .

If , then , which yields , and . Hence if .

For obtaining the result of this theorem in this case, the left work is to prove the method is T-stable for .

If , then , ; that is, Since , we get . Since we have from and , and therefore which means that the Heun method (12) is T-stable if .

(b) When , it is easy to know , and the Heun method (12) is T-stable if .

(c) When , if , then . If , then . Solving this gives , which contradicts the condition in (9). So if . Hence the Heun method in (12) is T-stable if .

The discussion above shows that the Heun method in (12) is T-stable provided that , which completes the proof.

Theorem 6. Suppose (7) meets the condition in (10). The Heun method in (13) is T-stable if , where

Proof. The condition in (10) gives . Denote and then from (17).

(a) Suppose , , , and . With , we have and from . With , we have ; that is, . The inequality with gives , and therefore . With this and , we get . Consequently, which means the Heun method in (13) is T-stable.

In the same way, we can prove that the Heun method in (13) is T-stable when , or , or , .

(b) Suppose , , and . With , we get and from . With , we have ; that is, . The inequality with gives , and therefore . With this and , we get . Consequently, which means that the Heun method in (13) is T-stable.

In the same way, we can prove that the Heun method in (13) is T-stable when , or , or , .

The proof of the theorem is complete.

4. T-Stability of the Balanced Method

The balanced method applied to (6) and (7), respectively, gives where and and are real numbers. In the analysis of T-stability of balanced method, we also use two-point random variables for the driving process.

For the balanced method in (31), we have Denote The discussion about the Heun method in Section 3 implies if , which means that the balanced method in (31) is T-stable if .

Similarly, for the balanced method in (32), we get and if , which means that the Heun method in (32) is T-stable if .

Theorem 7. Suppose (6) meets the condition in (9). The balanced method in (31) is T-stable if , where

Proof. The condition in (9) gives . Denote and from (34). It is easy to know for .

If , we have from the assumption , which applies . Then and in become Since , we have and which means that balanced method in (31) is T-stable.

In the same way, we can prove that the balanced method in (31) is T-stable when

The proof of the theorem is complete.

Theorem 8. Suppose (7) meets the condition in (10). The balanced method in (32) is T-stable if and , where

Proof. The condition in (10) gives . Denote and then from (35). It is easy to know for .

If , we have from and from the assumption , which yields . Then and in become Since , we have , and , which means that the balanced method in (32) is T-stable.

In the same way, we can prove that the balanced method in (32) is T-stable when

If , we get from and from the assumption , which applies . Then from . Since , we have consequently, , which means that the balanced method in (32) is T-stable.

In the same way, we can prove that the balanced method in (32) is T-stable when

The proof of Theorem 8 is complete.

5. Numerical Examples

Consider test equations (6) and (7) with . In the following figures s are nodes, and denotes the numerical solution at .

Take , and in (6). From (18), we get , which means that the Heun method in (12) is T-stable if . Figure 1 shows that the Heun method in (12) is T-stable when , , and but is unstable when , since exceeds the range of in Theorem 5, which verifies Theorem 5.

545830.fig.001
Figure 1: Simulations with the Heun method for (6) with , and . , , , and .

Take , and in (7). From (26), we get ,   , which means that the Heun method in (13) is T-stable if . Figure 2 shows that the Heun method in (13) is T-stable when , , and , but it is unstable when , since exceeds the range of in Theorem 6, which verifies Theorem 6.

545830.fig.002
Figure 2: Simulations with the Heun method for (7) with , and . , , , and .

Take , and in (6) and in the balanced method in (31). Then the balanced method in (31) is T-stable if from Theorem 7. Figure 3 shows that the balanced method in (31) is T-stable when , , , and , since these h’s are in the range of in Theorem 7, which verifies Theorem 7.

545830.fig.003
Figure 3: Simulations with the balanced method for (6) with , and . , , , and .

Take , , , and in (7) and , in the balanced method (32). Then the balanced method in (32) is T-stable if from Theorem 8. Figure 4 shows that the balanced method in (32) is T-stable when , , and , since these h’s are in the range of in Theorem 8, which verifies Theorem 8.

545830.fig.004
Figure 4: Simulations with the balanced method for (7) with , and . , , , and .

6. Conclusion

In this paper, T-stability of the Heun methods and the balanced methods for two kinds of linear SDDEs is studied. The Wiener increment of the numerical methods in this paper is approximated by a discrete random variable with two-point distribution in the process of the study of T-stability. The T-stable conditions for the Heun methods and balanced methods are given, respectively, and are verified by some numerical examples.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (no. 61272024) and the Anhui Provincial Natural Science Foundation (no. 11040606M06).

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