The Stochastic Θ-Method for Nonlinear Stochastic Volterra Integro-Differential Equations - Peng Hu; Chengming Huang (Abstract and Applied Analysis)
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.