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

Abstract EN

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.