Mean-square stability and convergence of a split-step theta method for stochastic Volterra integral equations

Abstract In this paper, a split-step θ method is introduced for solving stochastic Volterra integral equations with general smooth kernels. First, the mean-square boundedness and convergence properties of the numerical solution are analyzed. In particular, when the kernel function in the stochastic integral term satisfies a certain condition, the method can achieve superconvergence. Then, the mean-square stability of the method with respect to a convolution test equation is studied. A recurrence relation is found and mean-square stability regions are given by using root locus method. In particular, when the test equation degrades to the deterministic case, the new method with θ ≥ 1 ∕ 2 is V 0 -stable (i.e., it can preserve the stability of the convolution test equation), which is superior to the stochastic θ method. Finally, some numerical experiments are given to verify the theoretical results.

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