Mittag-Leffler Euler integrator for a stochastic fractional order equation with additive noise
A semilinear stochastic fractional order equation, and its deterministic counterpart, is considered. Full discretization of the model problem is carried out and optimal strong rate of convergence is proved, which is (almost) twice the rate of the rate for the implicit Euler method. A generalised exponential Euler method, named here as the Mittag-Leffler Euler integrator, is used for the temporal discretization. Spatial discretization by the spectral Galerkin method is then performed. The framework allows for nonlinearities from a general class of Nemytskij operators. Multiple spatial dimension is allowed when the noise is of trace class. Numerical experiments are presented to validate the theory.
stochastic differential equations.
strong conver- gence