The stochastic -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 -method is convergent of order 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 -method is mean-square asymptotically stable for every stepsize if and when , the stochastic -method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.
"The Stochastic -Method for Nonlinear Stochastic Volterra Integro-Differential Equations." Abstr. Appl. Anal. 2014 (SI66) 1 - 13, 2014. https://doi.org/10.1155/2014/583930