The fast Euler-Maruyama method for solving multiterm Caputo fractional stochastic delay integro-differential equations

This paper studies a type of multiterm fractional stochastic delay integro-differential equations (FSDIDEs). First, the Euler-Maruyama (EM) method is developed for solving the equations, and the strong convergence order of this method is obtained, which is \(\varvec{\min \left\{ \alpha _{l}-\frac{1}{2}, \alpha _{l}-\alpha _{l-1}\right\} }\) $\mathbf{min}\left\{{\mathit{\alpha }}_{\mathit{l}},\mathbf{-},\frac{\mathbf{1}}{\mathbf{2}},\mathbf{,},{\mathit{\alpha }}_{\mathit{l}},\mathbf{-},{\mathit{\alpha }}_{\mathit{l}\mathbf{-}\mathbf{1}}\right\}$ . Then, a fast EM method is also presented based on the exponential-sum-approximation with trapezoid rule to cut down the computational cost of the EM method. In the end, some concrete numerical experiments are used to substantiate these theoretical results and show the effectiveness of the fast method.