- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 157498, 9 pages
Strong Convergence of the Split-Step Theta Method for Stochastic Delay Differential Equations with Nonglobally Lipschitz Continuous Coefficients
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
Received 27 April 2014; Accepted 6 August 2014; Published 20 August 2014
Academic Editor: Jaeyoung Chung
Copyright © 2014 Chao Yue and Chengming Huang. 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.
This paper is concerned with the convergence analysis of numerical methods for stochastic delay differential equations. We consider the split-step theta method for nonlinear nonautonomous equations and prove the strong convergence of the numerical solution under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. In particular, these conditions admit that the diffusion coefficient is highly nonlinear. Furthermore, the obtained results are supported by numerical experiments.
Stochastic delay differential equations (SDDEs) play an important role in modeling some real-world phenomena in many scientific areas, such as economics , biology [2, 3], and medicine [4, 5]. However, many SDDEs arising in applications cannot be solved analytically; hence one needs to develop effective numerical methods to solve them.
In recent years, the numerical solution of SDDEs has attracted much attention and a number of numerical methods have been constructed (see, e.g., [6–8]). An important topic in this context is the investigation of the convergence of numerical methods and a number of interesting results have been found (see, e.g., [9–13]). In the analysis of strong convergence, a widely used assumption is that the drift and diffusion coefficients satisfy global Lipschitz and linear growth conditions [9–11]. In order to weaken this assumption, Mao and Sabanis  proved strong convergence of Euler-Maruyama type methods with local Lipschitz conditions and the bounded th moments () for solving SDDEs. Wang and Gan  showed that the improved split-step backward Euler method is convergent in the mean square sense under the condition that the diffusion coefficient is globally Lipschitz, and the drift coefficient satisfies a one-sided Lipschitz condition in the nondelay variable and a global Lipschitz condition in the delay variable . Bao and Yuan  proved the convergence rate of the Euler-Maruyama (EM) scheme for a class of SDDEs, where the corresponding coefficients may be highly nonlinear with respect to the delay variables. The strong convergence was also studied in [16, 17]. Nevertheless, all the above results are derived for SDDEs of which the diffusion coefficient with respect to the nondelay variables satisfies a linear growth or global Lipschitz condition. For example, they cannot be applied to some highly nonlinear problems such as
In this paper, we study the strong convergence of the split-step theta method  under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. These conditions admit that the diffusion coefficient with respect to the nondelay variables is highly nonlinear; that is, it does not necessarily satisfy a linear growth or global Lipschitz condition.
The structure of this paper is organized as follows. First, the existence of a unique solution of SDDEs under weaker conditions is recalled in Section 2. Then, some moment properties of the split-step theta method (3) are investigated in Section 3, while its strong convergence is derived in Section 4. Finally, some numerical results to support our theorems are presented.
2. Existence and Uniqueness of Solution
Throughout this paper, we denote both the Euclidean vector norm and the Frobenius matrix norm by and the complete probability space by . is increasing and continuous, and contains all -null sets. Let and . In this paper, we consider the numerical solution of the SDDEs in Itô’s sense: where and is -measurable, -valued random variable satisfying . Here denotes a standard -dimensional Brownian motion defined on the probability space.
Let be a time stepsize satisfying with a positive integer . Then the split-step theta method (SST) introduced in  for problem (2) reads where , is an approximation to , is a fixed parameter, , and for . For the given stepsize , let be the greatest integer satisfying . For simplicity, we directly assume in the following.
Huang  studied the exponential mean square stability of the SST method (3) under the following condition: where is a real symmetric, positive definite matrix. It is proved that when and , SST method (3) is exponentially mean square stable for all positive stepsizes. In this paper, we further study its strong convergence property under weaker conditions.
To ensure the existence of a unique solution on to SDDEs (2), we introduce the following assumption.
Assumption 1. The functions and in (2) are continuous in , , and and satisfy the nonglobal Lipschitz condition. More precisely, there exist positive constants , , , and such that where , , and .
Now, we state the following lemma which will play an important role in proving the strong convergence of the SST method (3).
Proof. Let . Then by using Itô formula, we infer that According to Assumption 1 and taking mathematical expectation on both sides of (10), we have Using the Gronwall inequality yields Applying Fatou’s lemma to (12), we obtain The proof is completed.
3. Moment Properties of SST
Assumption 3. There exists a positive constant , such that for and .
Proof. First, by Lemma 2, we know that is bounded. Denoting and then inserting (17) into (3), we have where ,,. By recursive calculation, we obtain Noting that , is independent of ; we have and . Hence, taking the mathematical expectation on both sides of (19) and then substituting (6) into (19) lead us to Next, using and regrouping (20), we have where . By virtue of recursive calculation from to , we know that is bounded. Applying the discrete Gronwall’s inequality, we have Therefore, there exists a positive constant , such that By using (3), (20), and (23), we infer Hence, by using Lemma 2, (23), and (24), there exists a positive constant which is independent of , such that The proof is completed.
4. Strong Convergence
In this paper, it is convenient to use continuous-time approximation solution. First, we denote and is the largest integer of . Then, we define continuous version of in (3) as follows: where . For convenience, rewriting (27) in integral form where , , and for . From (27) and (28), we easily verified that .
Assumption 5. There exist a positive constant and a positive integer , such that for and .
We now establish the following lemma, which will play a key role in proving the convergence of the SST method (3).
Proof. Let and . Then, by Itô formula, we derive Applying Assumption 1 and taking the mathematical expectation on both sides of (31) lead us to By using Lemma 4 and the Hölder inequality we have Next, we will bound the first term on the right-hand side of (33). According to Assumption 1 and Lemma 4, for , we have and then there exists a positive constant , such that where depends on , , , and . For , using (27), (35), and Lemma 4, we obtain that where is a positive constant which depends on . Substituting (35) and (37) into (33), we have Now, for any given , we choose such that for any Then, we can choose , such that for any Therefore .
We are now ready to prove the strong convergence of the SST method (3).
Proof. We divided our proof into three steps for readability. First, we define and . By applying Young’s inequality (, ), we obtain that for
Second, to bound the first term on the right-hand side of (43), it is enough to show that is bounded due to the Lyapunov inequality. By using Burkholder-Davis-Gundy inequality, we obtain Furthermore, using Assumption 5, Lemma 4, Assumption 1, and (37), we get where is a positive constant dependent of . Similarly, we have By substituting (45) and (46) into (44), we obtain Applying the Gronwall inequality and the Lyapunov inequality leads us to and then Finally, combining (49) and (43), we have For any given , by Lemmas 2 and 4, we can choose such that Now, by (8) and (39), there exists such that for Furthermore, by Lemma 6, we choose sufficiently small such that The proof is completed.
5. Numerical Results
In this section we consider the following numerical experiments that confirm the conclusions obtained in the previous sections.
For the first example, we consider the nonautonomous SDDEs with initial value , .
For the second example, we consider the SDDEs  with initial value , .
It is easy to show that SDDEs (54) and (55) satisfy Assumptions 1, 3, and 5. Following the idea of  and denoting by the numerical approximation to at end point in the th simulation of all simulations, we approximate means of absolute errors at terminal time by
It is difficult to obtain the analytic form of the exact solution to SDDEs (54) and (55). Recall that Theorem 7 guarantees that the SST method (3) strongly converges to the exact solution. Therefore, it is reasonable to identify numerical solution obtained by the SST method (3) () using the very small stepsize as the “exact” solution. With the “exact” solution at hand, we can follow to obtain numerical solution by the SST method (3) using different stepsizes , , , , on the same discretized path. We generate different discretized Brownian paths over and apply the formula (56) to obtain the absolute errors. Errors at for the SST method (3) solving SDDEs (54) and (55) with different stepsizes are listed in Tables 1 and 2, respectively. In Figure 1, we plot the means of absolute errors against on a log-log scale.
In this work, we carried out a strong convergence analysis on the SST method for SDDEs under a local Lipschitz condition and a coupled condition on the drift and diffusion coefficients. Different from most of the existing convergence results for SDDEs, our results can be applied to equations of which the diffusion coefficient with respect to the nondelay variables is highly nonlinear. Both theoretical analysis and numerical tests show that the SST method is efficient for the numerical solution of SDDEs.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the referees and the editors for their valuable comments and suggestions. This work was supported by NSF of China (nos. 91130003 and 11371157) and the Fundamental Research Funds for the Central Universities (no. 2013TS137).
- M.-K. Lee, J.-H. Kim, and J. Kim, “A delay financial model with stochastic volatility; Martingale method,” Physica A: Statistical Mechanics and Its Applications, vol. 390, no. 16, pp. 2909–2919, 2011.
- X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited,, Chichester, UK, Second edition, 2008.
- T. Tian, K. Burrage, and P. M. a. Burrage, “Stochastic delay differential equations for genetic regulatory networks,” Journal of Computational and Applied Mathematics, vol. 205, no. 2, pp. 696–707, 2007.
- M. Carletti, “Numerical solution of stochastic differential problems in the biosciences,” Journal of Computational and Applied Mathematics, vol. 185, no. 2, pp. 422–440, 2006.
- A. F. Ivanov and A. V. Swishchuk, “Optimal control of stochastic differential delay equations with application in economics,” International Journal of Qualitative Theory of Differential Equations and Applications, vol. 2, no. 2, pp. 201–213, 2008.
- 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.
- Y. Hu, S.-E. A. Mohammed, and F. Yan, “Discrete-time approximations of stochastic delay equations: the milstein scheme,” The Annals of Probability, vol. 32, no. 1, pp. 265–314, 2004.
- C. Huang, “Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations,” Journal of Computational and Applied Mathematics, vol. 259, pp. 77–86, 2014.
- Z. Fan, M. Liu, and W. Cao, “Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1142–1159, 2007.
- 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.
- H. Zhang, S. Gan, and L. Hu, “The split-step backward Euler method for linear stochastic delay differential equations,” Journal of Computational and Applied Mathematics, vol. 225, no. 2, pp. 558–568, 2009.
- X. Wang and S. Gan, “The improved split-step backward Euler method for stochastic differential delay equations,” International Journal of Computer Mathematics, vol. 88, no. 11, pp. 2359–2378, 2011.
- 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.
- 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.
- J. Bao and C. Yuan, “Convergence rate of EM scheme for SDDEs,” Proceedings of the American Mathematical Society, vol. 141, no. 9, pp. 3231–3243, 2013.
- F. Wu, X. Mao, and K. Chen, “The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2641–2658, 2009.
- J. Liu and J. Zhou, “Convergence rate of Euler-Maruyama scheme for stochastic pantograph differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 6, pp. 1697–1705, 2014.
- X. Mao and M. J. Rassias, “Khasminskii-type theorems for stochastic differential delay equations,” Stochastic Analysis and Applications, vol. 23, no. 5, pp. 1045–1069, 2005.
- D. J. Higham, X. Mao, and A. M. Stuart, “Strong convergence of Euler-type methods for nonlinear stochastic differential equations,” SIAM Journal on Numerical Analysis, vol. 40, no. 3, pp. 1041–1063, 2002.
- D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001.