Strong superconvergence of the Euler–Maruyama method for linear stochastic Volterra integral equations

Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion

“…Moreover, the related conclusions for the other classes of SVIEs by using Euler-type method can be obtained [20,31,32]. Similarly, if we add a jump term to the right side of the SVIEs, Khalaf et al [33] showed that the strong convergence order can reach up to order 1 if the kernel function is Lipschitz continuous and the diffusion coeffcient and the jump coeffcient satisfy a same additional assumption as in [30]. For the SVIEs with doubly singular kernels, Dai and Xiao [34] analysed the strong convergence order of EM method, and constructed the fast EM method to improve the computational effciency.…”

“…Indeed, for the strong convergence order of numerical methods for SVIEs, some interesting conclusions also have been obtained. For the linear SVIEs with convolution kernels, Liang et al [30] obtained the superconvergence order of EM method when the kernel function is Lipschitz continuous and satisfies an additional assumption. Moreover, the related conclusions for the other classes of SVIEs by using Euler-type method can be obtained [20,31,32].…”

Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation

“…Under the same settings with the EM approximation (5), we can modify it by the left rectangle rule [21,26] as…”

A sharp error estimate of Euler‐Maruyama method for stochastic Volterra integral equations

“…In recent years, some important progresses for the numerical algorithm of the SVIE (1.1) with 𝜎 = 0 has been proposed, for example, Galerkin methods, 25 block pulse approximation, [26][27][28] spectral collocation method, [29][30][31] operational matrix method, [32][33][34] improved rectangular method, [35][36][37] B-spline collocation method, [38][39][40] Milstein method, 41,42 meshless discrete collocation method, 43 and Euler-Maruyama (EM) method. [44][45][46][47] In this paper, we are interested in numerically solving the nonlinear SVIE (1.1) for which 𝜎 > 0 by using EM method and devoting to two main goals.…”