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

Abstract

The stochastic $\mathrm{\Theta }$-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 $\mathrm{\Theta }$-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 $\mathrm{\Theta }$-method is mean-square asymptotically stable for every stepsize if $1/2\le \theta \le 1$ and when , the stochastic $\mathrm{\Theta }$-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.