An Euler-Maruyama method and its fast implementation for multiterm fractional stochastic differential equations

In this paper, we derive an Euler-Maruyama (EM) method for a class of multiterm fractional stochastic nonlinear differential equations and prove its strong convergence. The strong convergence order of this EM method is min{alpha m-0.5,alpha m-alpha m-1}$$ \min \left\{{\alpha}_m-0.5,{\alpha}_m-{\alpha}_{m-1}\right\} $$, where {alpha i}i=1m$$ {\left\{{\alpha}_i\right\}}_{i equal to 1} circumflex m $$ is the order of Caputo fractional derivative satisfying that 1>alpha m>alpha m-1>MIDLINE HORIZONTAL ELLIPSIS>alpha 2>alpha 1>0,alpha m>0.5$$ 1>{\alpha}_m>{\alpha}_{m-1}>\cdots >{\alpha}_2>{\alpha}_1>0,{\alpha}_m>0.5 $$, and alpha m+alpha m-1>1$$ {\alpha}_m+{\alpha}_{m-1}>1 $$. Then, a fast implementation of this proposed EM method is also presented based on the sum-of-exponentials approximation technique. Finally, some numerical experiments are given to verify the theoretical results and computational efficiency of our EM method.