Well-posedness and EM approximation for nonlinear stochastic fractional integro-differential equations with weakly singular kernels.

This paper considers the initial value problems of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels under the local Lipschitz condition. First of all, the existence, uniqueness and stability results of the problems under studying are derived in detail. Second, the modified Euler-Maruyama approximation is presented for solving numerically the equation, and then its strong convergence is proven. Moreover, we investigate the convergence rate of this method to show its computational efficiency. Finally, several numerical tests are reported for verification of the theoretical findings.