The stochastic \(\Theta\)-method for nonlinear stochastic Volterra integro-differential equations
Summary: The stochastic \(\Theta\)-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 \(\Theta\)-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 \(\Theta\)-method is mean-square asymptotically stable for every stepsize if \(1 / 2 \leq \theta \leq 1\) and when \(0 \leq \theta < 1 / 2\), the stochastic \(\Theta\)-method is mean-square asymptotically stable for some small stepsizes. Finally, we validate our conclusions by numerical experiments.
scientific article; zbMATH DE number 7022655