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

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  • Peng Hu
  • Chengming Huang

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

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Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:583930
DOI: 10.1155/2014/583930
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