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Volume 8, Issue 2
Mean Square Convergence of Stochastic $\theta$-Methods for Nonlinear Neutral Stochastic Differential Delay Equations

S. Gan, H. Schurz & H. Zhang

Int. J. Numer. Anal. Mod., 8 (2011), pp. 201-213.

Published online: 2011-08

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  • Abstract

This paper is devoted to the convergence analysis of stochastic $\theta$-methods for nonlinear neutral stochastic differential delay equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values, which leads to a stochastic differential algebraic system. Drift-implicit stochastic $\theta$-methods are proposed for the coupled system. It is shown that the stochastic $\theta$-methods are mean-square convergent with order 1/2 for Lipschitz continuous coefficients of underlying NSDDEs. A nonlinear numerical example illustrates the theoretical results.

  • Keywords

neutral stochastic differential delay equations, mean-square continuity, stochastic theta-methods, mean-square convergence, consistency.

  • AMS Subject Headings

65C30, 60H10, 60H35, 65C20

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{IJNAM-8-201, author = {S. Gan, H. Schurz and H. Zhang}, title = {Mean Square Convergence of Stochastic $\theta$-Methods for Nonlinear Neutral Stochastic Differential Delay Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2011}, volume = {8}, number = {2}, pages = {201--213}, abstract = {

This paper is devoted to the convergence analysis of stochastic $\theta$-methods for nonlinear neutral stochastic differential delay equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values, which leads to a stochastic differential algebraic system. Drift-implicit stochastic $\theta$-methods are proposed for the coupled system. It is shown that the stochastic $\theta$-methods are mean-square convergent with order 1/2 for Lipschitz continuous coefficients of underlying NSDDEs. A nonlinear numerical example illustrates the theoretical results.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/682.html} }
TY - JOUR T1 - Mean Square Convergence of Stochastic $\theta$-Methods for Nonlinear Neutral Stochastic Differential Delay Equations AU - S. Gan, H. Schurz & H. Zhang JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 201 EP - 213 PY - 2011 DA - 2011/08 SN - 8 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/682.html KW - neutral stochastic differential delay equations, mean-square continuity, stochastic theta-methods, mean-square convergence, consistency. AB -

This paper is devoted to the convergence analysis of stochastic $\theta$-methods for nonlinear neutral stochastic differential delay equations (NSDDEs) in Itô sense. The basic idea is to reformulate the original problem eliminating the dependence on the differentiation of the solution in the past values, which leads to a stochastic differential algebraic system. Drift-implicit stochastic $\theta$-methods are proposed for the coupled system. It is shown that the stochastic $\theta$-methods are mean-square convergent with order 1/2 for Lipschitz continuous coefficients of underlying NSDDEs. A nonlinear numerical example illustrates the theoretical results.

S. Gan, H. Schurz and H. Zhang. (2011). Mean Square Convergence of Stochastic $\theta$-Methods for Nonlinear Neutral Stochastic Differential Delay Equations. International Journal of Numerical Analysis and Modeling. 8 (2). 201-213. doi:
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