Abstract and Applied Analysis

The Stochastic Θ -Method for Nonlinear Stochastic Volterra Integro-Differential Equations

Peng Hu and Chengming Huang

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Abstract

The stochastic Θ -method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochastic Θ -method is convergent of order 1 / 2 in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochastic Θ -method is mean-square asymptotically stable for every stepsize if 1 / 2 θ 1 and when 0 θ < 1 / 2 , the stochastic Θ -method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 583930, 13 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425048218

Digital Object Identifier
doi:10.1155/2014/583930

Mathematical Reviews number (MathSciNet)
MR3275749

Citation

Hu, Peng; Huang, Chengming. The Stochastic $\mathrm{\Theta }$ -Method for Nonlinear Stochastic Volterra Integro-Differential Equations. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 583930, 13 pages. doi:10.1155/2014/583930. https://projecteuclid.org/euclid.aaa/1425048218


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