Milstein Scheme for a Stochastic Semilinear Subdiffusion Equation Driven by Fractionally Integrated Multiplicative Noise

This paper investigates the strong convergence of a Milstein scheme for a stochastic semilinear subdiffusion equation driven by fractionally integrated multiplicative noise. The existence and uniqueness of the mild solution are established via the Banach fixed point theorem. Temporal and spatial regularity properties of the mild solution are derived using the semigroup approach. For spatial discretization, the standard Galerkin finite element method is employed, while the Grünwald–Letnikov method is used for time discretization. The Milstein scheme is utilized to approximate the multiplicative noise. For sufficiently smooth noise, the proposed scheme achieves the temporal strong convergence order of O(τα), α∈(0,1). Numerical experiments are presented to verify that the computational results are consistent with the theoretical predictions.