## Abstract and Applied Analysis

### Almost Sure and ${L}^{p}$ Convergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation

#### Abstract

The convergence of the split-step backward Euler (SSBE) method applied to stochastic differential equation with variable delay is proven in ${L}^{p}$-sense. Almost sure convergence is derived from the ${L}^{p}$ convergence by Chebyshev’s inequality and the Borel-Cantelli lemma; meanwhile, the convergence rate is obtained.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 390418, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412277037

Digital Object Identifier
doi:10.1155/2014/390418

Mathematical Reviews number (MathSciNet)
MR3219367

Zentralblatt MATH identifier
07022289

#### Citation

Guo, Qian; Tao, Xueyin. Almost Sure and ${L}^{p}$ Convergence of Split-Step Backward Euler Method for Stochastic Delay Differential Equation. Abstr. Appl. Anal. 2014 (2014), Article ID 390418, 7 pages. doi:10.1155/2014/390418. https://projecteuclid.org/euclid.aaa/1412277037

#### References

• 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.
• 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.
• Y. Hu, S.-E. 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.
• 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.
• 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.
• 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.
• I. Gyöngy and S. Sabanis, “A note on Euler approximations for stochastic differential equations with delay,” Applied Mathematics and Optimization, vol. 68, no. 3, pp. 391–412, 2013.
• S. Janković and D. Ilić, “An analytic approximation of solutions of stochastic differential equations,” Computers & Mathematics with Applications, vol. 47, no. 6-7, pp. 903–912, 2004.
• X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing Limited, Chichester, UK, 2nd edition, 2007. \endinput